Since taking lessons is a very expensive thing these days, I'd like to know some tips on how to extend the range of the voice and maybe also on how to switch from one register to another without hearing that bad difference between one and the other.
Hell, always worth trying out a Chris Cornell: belt it until you can sing it without breaking.
Of course, if you don't want to butcher your voice, try sirens. Taking it as long as is comfortable, and back up, and then up to the top as far as is comfortable, and back down, just a little a day will get your voice used to the exercise.
To extend the range of your voice you need to practise on "both sides". The higher notes and the deeper notes. I made the expierience the more I practised deeper notes the higher I got over time.
Here's how you could practise deeper notes: Try to talk everyday in a deep voice for some time. Note: if it hurts you're doing it wrong. Don't exaggerate this. Find a note that makes you feel comfortable and try to keep talking in that voice. Also buzz along with songs on the radio with deep notes. Important is also your breath control. I don't want to go in detail here, because breath control is a really big topic.
Well, don't expect to extend your range beyond what you can reach now. What exercise does is mostly stabilizing the range, giving solid control over it, and making it worthwhile hearing. But the actual range with which you can produce sort of noise right now is not all that likely to change.
Regarding the break, it very much depends on what kind of music or style you are singing and whether you are doing cover music or writing music for your own voice.
If you are ending up in a more falsetto/head voice based style, the basic piece of work is practising to sing through the break. There are patterns one can use for that, like trying to sing the legato patterns g f e d c, g e f d e c, g f g e g d g c, each pattern taking about a second not counting its final note. You do the exercise on different vowels and start the patterns where the top note is barely in falsetto (different vowels may have different pitches for the break, though), then shift the exercise up a semitone after doing it a few times. The basic thing is you have to just not care going through the break. Pitch control, breathing and other stuff change, and you want those changes to happen reflexively. But mouth position and everything affecting the resonances and airways should stay.
Another thing to realize is that the basic action in the break is a change in the configuration of your larynx: it's basically a mechanical switch making the vocal fold's edges drawn taut from different control points. It takes years of practice to make this spring-controlled switch not just flip over but move in a controlled manner. But the point is that the largest principal changes are the fixtures and consequently the mechanical resonances in your head.
By far the largest audible change happens for yourself because of that. For the listener, the break is much less noticeable. So you need to learn to just ignore it and sing over and through it. The break itself is much less obvious to the listener than the loss of control while you adapt to the flipping of the switch. And that can be practiced even long before you learn to influence the nature of the flip itself (don't wait for it, don't bank on it).