Recently I've been wanting to play electric guitar, because it just sounds better than classical or acoustic, for the songs I want to play at least. I already know the basics and probably more, since I learned to play classical guitar for 6 years. (I'm 16, f btw) What I'm mostly interested in is the transition and gear: what's best to get started on, is it very different, is it better if I buy something cheap at first or really invest in it. I don't personally know anyone who plays electric guitar to help me with this. Also should I find a teacher or someone? I don't have much knowledge, but I have enthusiasm.
"What's best to get started on? "
It seems like a good idea to begin with a versatile guitar. Strat-style guitars with three single-coil pickups are very versatile, easy to play, easy to intonate, and hold their tunings well. Gibson-style guitars, i.e., Les Pauls, ES 335s, etc, have two humbucking pickups, and a (typically) shorter scale-length. Humbuckers have a distinctive sound, but these can be pretty versatile too. The shorter scale-length means that these guitars are a little harder to intonate and don't hold their tunings quite as well as guitars with longer scale-lengths. If you like the warm sound of humbuckers, I would suggest getting a guitar with coil-tap switches so that you can select single-coil modes for more versatility.
In any case, it would be best to go to some music shops to play on as many electric guitars as possible before buying one; some will feel better than others to you. When you find one you like, have the store guitar tech check the intonation and string action (and make any necessary adjustments) before you buy it.
"Is it better if I buy something cheap at first or really invest in it?"
I would suggest that most beginners should get an inexpensive guitar and amp to start, but since you already have been playing for 6 years, demonstrating some level of commitment, you could consider a better instrument if the funds are there. In any case, don't buy an instrument that has obvious defects or that plays poorly. You don't have to pay a lot of money for a very decent guitar to learn with, and there is no need to spend a lot when you don't know exactly what you will need in a few years. You can always trade up to a better instrument.
"Is it very different?"
String-spacings are typically quite a bit wider on classical guitars than on electric or steel-string acoustic guitars, and necks on electric guitars are often a bit slimmer than necks on classical guitars. Fender-style necks tend to be slimmer than Gibson-style necks; you might be aware of this when you are trying out different instruments to see what feels best.
The mechanics of play are more-or-less the same, but you will probably be playing with a plectrum on the electric guitar. Picking technique may be a new thing that you will need to work on.
Adapting to new styles of play will likely be one of the biggest challenges. How to comp and fit in with other players? How to integrate chords and melody? How to play compelling melodies? How to improvise melodies? How to improvise accompaniments? All players have to wrestle with these things, but it is easy to take for granted what you already know in styles that you are comfortable in; when you try a new style, these questions need to be addressed anew.
"Should I find a teacher or someone?"
Yes. But this is not a requirement; you may do perfectly well on your own, especially given that you have several years of classical lessons behind you. Still, a good teacher can help you identify problems, make technical adjustments, and help you with new techniques (like picking technique). A good teacher can also help you answer some of the stylistic questions mentioned in the last paragraph.
It would also be beneficial to find ways to see live music that interests you. Watch the players that you like and try to figure out what they are doing. If you can, talk with some of these players. You can pick up all kinds of useful information by talking with serious players.
(In addition to David Bowling's excellent answer, I'd like to give a few practical and specific suggestions. Some things will obviously overlap with his answer.)
Choosing a type of guitar
When buying a guitar as a beginner, especially if you're switching from classical guitar, feel and playability are more important than details of tone. Especially the width, thickness, curvature and scale length of the neck will be much more important than the shape of the body, the number and type of pickups, or the type of wood used for the various parts of the guitar. So I would suggest you go to a guitar store and play several guitars for a few minutes to get an idea of how they feel. You don't even have to plug them into an amp, just play something that you're used to playing on your classical guitar, and see how you get on.
Most electric guitars you will find in your local guitar shop will be based on a handful of classic designs from the 1950's and 60's, like the Fender Stratocaster, the Fender Telecaster, and the Gibson Les Paul. If you can try out only 3 guitars, try to play a Fender (or its subsidiary Squier) Stratocaster or Telecaster or Jazzmaster (these have a 25.5" scale thin neck), a Gibson (or its subsidiary Epiphone) Les Paul or SG (these have a very different shape and feel and a shorter 24.75" scale neck), and a Fender (or Squier) Jaguar or Mustang (these have a short 24" scale). The neck width decides the distance between the strings, and the difference in scale length not only means that the distance between the frets is different, but also that the tension on the strings is different.
Additionally, you could try a semi-acoustic hollow-body guitar, like a Gibson ES-335 or an Epiphone copy of a Gibson or Gretsch model.
Choosing an affordable guitar
Once you've decided which guitar model works best for you, there is a wide range of brands, qualities and prices to choose from. As mentioned, both Fender and Gibson have subsidiary companies that make cheaper versions of their well-known models; for Fender this is Squier, for Gibson this is Epiphone. But many other companies make guitars in these styles.
If you're interested in a Stratocaster or "ST-type" style guitar, the obvious choice is Squier. However, for someone who already has had 6 years of guitar lessons, I would avoid the cheapest "Bullet" range, and go for the "Affinity" range, or preferably the "Standard" or "Vintage Modified" range. These come in the classic version with 3 single coil pickups, but also in a version with a humbucker and 2 single-coil pickups, which may be interesting if you want to use both clean and heavy distorted rock sounds.
A good alternative for the Squier Stratocaster is the Yamaha Pacifica. Again, in your case I would avoid the cheaper "012" range and go for the "112" range. Both the "112J" and "112V" models have a humbucker and two single-coil pickups, but the "112V" model comes with "coil splitting", which means you can use one half of the humbucker as a single-coil pickup, for greater versatility.
For a Telecaster or "T-type" style guitar, what I wrote about Squier Stratocasters is good advice here too: avoid the "Bullet" range and preferable also the "Affinity" range. You can get Squire Telecasters with the standard 2 single-coil pickups, or also with 2 humbuckers, depending on the style of music you want to play.
A good alternative for the Squier Telecaster with single-coil pickups is the Ibanez Talman TM302, or the TM303 which adds a third single-coil pickup.
The shorter-scale Jaguar and Mustang and also the Jazzmaster are very much their own unique thing, with few imitations from other brands. So again I'd suggest a mid-price Squier version.
For an affordable Gibson Les Paul or SG style guitar, Epiphone are quite unbeatable. They have many different ranges and the naming conventions are somewhat confusing, so it's difficult to give general advice, but again I would avoid the cheapest range, and you should be able to buy a decent Epiphone LP or SG for slightly more than a decent Squier. A semi-acoustic hollow-body will unfortunately cost around 50% more.
If you want to play any genre of metal, the look and sound of Ibanez or Jackson guitars will probably appeal to you. However, if you find other brands more comfortable to play, be assured that you can get the heaviest of rock sound out of almost any guitar with humbuckers, even if it doesn't look very "metal".
Some guitar models come with a "tremolo" system, to bend the pitch up and down by pressing a handle attached to the bridge. These are fun to play with, but make it harder to keep the guitar in tune, especially on cheaper guitars; I'd advise you to only buy a guitar with a tremolo if the style of music you want to play requires it.
Buying the guitar
If you buy a new guitar in a shop, tell the salesperson that you're switching from classical guitar, and what style of music you want to play, and let them advise you on what strings to put on it and how to set it up correctly, or have them do it for you.
If you want to buy a second-hand guitar, try to take someone with you who has more experience with electric guitars, otherwise you won't know what to look out for, and someone may try to sell you a terrible guitar that looks okay to a beginner.
Tube (or "Valve") amplifiers will probably be above your budget, but the additional sound quality they offer isn't really relevant for a beginner. A decent transistor amp for home and practice use can be really affordable. However, some of the really small amps you see offered as a package with beginner guitars can be of dubious quality; a good example is Fender: their cheapest range, the "Frontman" isn't worth much, but one step up, the "Champion 20", sounds decent and is very versatile. Most amp brands, like Vox, Marshall, Orange or Blackstar, make similarly priced small amps of comparable quality.
Recent advances in digital audio technology have led to a class of "modelling" amplifiers that are able to replicate the sound of many different amplifiers and effect pedals. These amps are extremely versatile, sound convincing and have become surprisingly affordable. The obvious example is the Boss Katana range. For someone in your situation, there's no reason to buy any amp that costs more than a Boss Katana 50; it's all you could ever want as a beginning electric guitar player. (Although it gives you so many sound options that you may get lost tweaking your sound instead of practicing your playing; if you think there's a chance of this happening, get something simpler.)
About effect pedals
As mentioned, many affordable amplifiers now come with at least a few built-in effects, like reverb, delay, chorus, tremolo... and offer different "crunch" or "drive" sounds with varying levels of distortion. As a beginner, you should have all the sounds you need to practice any style of music (except maybe the more extreme metal genres) right there, without the need for additional effect pedals. Again, as a beginner, buying effect pedals may cause to you concentrate on tweaking your sound instead of practicing your playing, so leave this for later.
The only possible exception would be a looper pedal. These allow you to record a few bars of your playing while you're playing, and then play it back in a loop so that you can use it as an accompaniment and play over it. This is great for practicing many aspects of your playing, and prepares you for playing together with other musicians.
If you can't find a teacher, or can't afford one, there is always the internet; there are so many video tutorials online, explaining every possible aspect of guitar technique and musical theory, that there's bound to be something that fits your technical level and musical taste.
NOTE: You'll notice that I didn't mention any famous players to describe the guitars. That's because playing like your favourite guitar player, sounding like them, and looking like them, are very different things. A guitar that works for them may not work for you, and a guitar that you associate with a particular genre may work well for you in a totally different genre. Telecaster were initially favoured by country musicians, but in the 60's and 70's they became the signature sound of some iconic rock musicians. Stratocasters were initially associated with the clean tones of 60's instrumentals, but were later used by some of the noisiest artists you'll ever hear. The Les Paul is synonymous with 70's hard rock, but is also favoured by many jazz guitarists. And the Jazzmaster was never really popular with jazz musicians, but has made a comeback thanks to a few alternative rock bands from the 90's. So give each guitar you try a chance, and don't limit yourself to what you think looks cool.
My suggestion would be to take some time before you spend money and go to a number of guitar stores and try a number of different guitars and amps. You may find one you absolutely love, if you do, keep looking to see if you can out do yourself and find one you love even more. Maybe you will and maybe you won't, but at least you'll know you tried. The quest for the right guitar never ends for some people and the same can be said for the right amp. A lot depends on what kind of music you wish to play and the size of room you'll be playing in, and even more depends on your ears and what you wish to be hearing. These days, there is something for almost everyone, but you've got to have an idea of what it is that trips your trigger. No one else can decide that for you.
Rather than stick with the "cheap is good for a first guitar" advice, I'd mention that one reason for starting off on a higher quality instrument is resale value, if it's not something you persevere with. Or if you decide to sell it because a different style of guitar suits you better.
Apparently there are parts of the world where pawn shops are filled with lightly used low-end Stratocasters sold by people who didn't persevere. Good for you if you want to buy second hand as you may get a bargain (which is also often-given advice). Not so good if you're trying to sell a low-end Stratocaster that you've decided you won't stick with...
Also, more expensive guitars are generally better quality (up to the level of diminishing returns for your money), more likely to hold a setup well (compared to many low budget guitars), and just all-round nicer to play. Therefore you'd be more likely to pick it up and play it, and enjoy it. And they hold their value better if you should come to sell it.
All very subjective of course: an expensive guitar can be a dog and a cheap one can handle beautifully.