Background: I'm a guitarist of 27 years who had piano lessons as a kid and got addicted to synthesizers about ten years ago. Now I have a whole suite of virtual synth plugins as well as a huge rack of physical synthesizers.
When a guitarist is working on their tone, they invest in technique, stompboxes, amps and so on. What does the modern metal producer do? Learn how to use additive synthesis to create the sound they want from the ground up? Buy lots of physical keyboards with good sounds? Plugins? Orchestral sample packs, or some other kind? Some combination of the above?
Kind of like guitarists, everyone has their own approach to synthesis, producing, and music in general. I do want to say that subtractive synthesis is much more famous and more popular than additive synthesis. Most synths you're hearing across all genres are subtractive.
I'd like to learn a bit about pros and cons and common practices before investing loads of money and time in learning new things and buying gear. I'm not asking what specific products to buy, but what techniques are generally used.
Jumping off of this paragraph and Tetsujin's excellent answer, I have a suggestion for what you might think about in terms of getting into synthesis: Think about how you approach guitar and use that as a basis for exploring synths.
A synth is just another instrument. Everyone has their favorite kind of guitar, pickups, effects, analog versus digital opinion, amps, cables, strings, etc. etc. Synths have a lot of analogous choices that come down to taste and opinion for the musicians. You can see that Tetsujin and I have different tastes when it comes to hardware versus virtual synths, and that's just the tip of the iceberg. In both the hardware and virtual synth worlds, there are a huge number of choices you can make and preferences you can have.
So I suggest approaching it similar to guitar. Maybe when you bought your first guitar, you either heard a sound and wanted to get that gear, or you just bought the one you could afford, or you bought the one that looked coolest to you, etc. Either way you probably had a limited budget and likely didn't get a '59 Les Paul as your first axe. But the first one you bought was something you could learn on and get the hang of it and develop your ear so you could start learning what you look for in a guitar. And likewise with amps and effects, etc.
So, do the same with synths! Either look for something affordable, pick a band/artist/producer to emulate and see what they use, play around with a bunch in your price range in a store or demoing software or watch YouTube videos and start with something that grabs you by the ear.
One way you can get a head start is to make the analog/digital/hardware/virtual decision based on how you are with guitars. If you've got a Line 6 Helix and the Helix plugin on your computer and you love those and really appreciate being able to take your laptop and an interface anywhere and get your guitar sound, then you'll probably prefer the power, flexibility, and portability of virtual synth plugins and a decent controller. On the other end of the spectrum, if you can't stand to play through anything other than your vintage Fender Deluxe Reverb with NOS tubes and Variac to imitate old-school wall power, then you'll probably be happier in the end with a 100% analog hardware synth (and there are some awesome affordable choices in that space these days).
I think a big part of your question might be how people chase the sounds they want on synths. With a synth, tone generation is a bit more flexible than on guitar. Whichever way you go on the virtual/hardware front, I strongly suggest looking to get a real or virtual subtractive analog synth. Such a synth or plugin would be based around 2 or 3 oscillators, a low pass filter, two envelope generators, and a voltage controlled amplifier (VCA). A low frequency oscillator (LFO) and some other things are often included, depending on the synth/plugin.
It's a bit of a journey, but with just those components (real or virtual), you can make all kinds of sounds from mellow to metal, and everything in between. Just like with guitar, synths are often used with effects, and many hardware and virtual synths include effects built-in. You can always tack on extra effects either when performing/recording or later on during mixing/producing. You can plug hardware synths right into your guitar rig and play through your pedalboard and amp! A lot of synths have been used that way over the years - the theremin solo in "Whole Lotta Love" is a classic example. So if you want your synth to sound more metal, add distortion! Or run it through a Dual Rectifier (real, modeled, or profiled). More ethereal - add compression and/or chorus and/or reverb. Your knowledge of guitar effects transfers almost 100% intact.
Obviously I could go on way too long. I'm going to cut myself off and include some places I suggest you look next to get into synths. Warning: this could end up consuming a lot of time and money!
- Automatic Gainsay is a YouTube channel by Marc Doty (who knows tons about synths) where he posts these excessively long, overly in-depth reviews and analyses of hardware synths. You can learn a lot from him. One video you might watch to understand the archetypal synthesizer (the Strat of synths, one might say) is his overview of the Moog Minimoog Model D
- Arturia is probably the closest thing the synth world has to a company like Line 6, but with a twist. Arturia got famous by making some great virtual synth plugins, and they are still one of the top names. They have a (fairly) affordable suite of plugins that includes pretty solid reproductions of (I have to go count) over 25 classic keyboard instruments, most of them synthesizers. But, Arturia also makes some amazing analog hardware synthesizers. And they make great controllers and some other awesome tools besides. No matter how you begin your synth journey, I highly recommend you look through the different options that Arturia offers.
- Native Instruments is probably king of the hardware/software integration world for virtual instruments and samples right now. If you really want tight integration between a USB controller keyboard and virtual instruments, my understanding is no one does it better (I've never used anything NI ever, so I can't say myself). You've definitely heard NI's flagship virtual synth, called Massive, on many recordings - soundtracks in particular.
- Moog Music in my opinion is very much like a synth version of Gibson USA. They are venerated as having made the first really portable powerful analog synth. Their products generally have excellent sound quality. Many consider their stuff to be overpriced. And they are not so wise when it comes to the business side of things (I tried to link to their home page but I think it's down or having issues?). If you're certain you want a hardware analog synth, you gotta check out Moog. I would look at their affordable options (the most affordable of which sadly require an external keyboard to play them), possibly consider the Grandmother if you want to jump into the deeper end (it's USD 999 but you'll never outgrow it as long as you keep using synths), and definitely check out the used market for the Moog Little Phatty (discontinued but generally affordable and awesome) and other options.
- Korg has of course been a huge name in digital synthesizers and keyboards since the 1980s, and they are still one of the top brands in that space. But they have put out some very nice and affordable analog synthesizers in recent years (check out the Monologue, the Minilogue, and the ARP Odyssey). They also have analog modeling synths, which are kind of like virtual instruments in a box with a keyboard - generally similar sound to analog at a lower price with more powerful features.
- Sequential Circuits AKA Dave Smith Instruments has nothing affordable and nothing for a beginner in their product lines, but I'm including them for completeness. They might be more like the Fender of the synth world compared to Moog. They made the first viable, portable polysynth back in the 1970s and like the Minimoog, you've heard the Prophet 5 and its descendants thousands of times over the decades since.
- If you want to go insane right out of the gate, Eurorack is not a company but a (loose) standard form factor and power supply system for combining individual synth modules into larger systems. Eurorack is more popular with the EDM, sequencing, and producing crowds because adding a keyboard and real-time performance control to Eurorack rigs can be a bit of a pain. If you're the kind of guitarist who likes to build an axe from scratch using Warmoth or Allparts or something like that, you might enjoy putting together Eurorack modules. The cost of entry seems modest, but it really isn't. Better to drop the money on a Moog Grandmother and get an all in one ready-to-go unit that sounds great than start down the Eurorack rabbit hole, in my personal opinion.