+1 for hooktheory. That site is incredibly helpful for theory, and the library of songs they get is ever expanding. They also have a theory tutorial somewhere on there, so don't be afraid to try that out.

I agree with a lot of the sentiment already expressed in this thread. I find them to be a useful tool, but I would advise to not lean on them as a crutch, if you wish to really master crafting your own sounds. With that said, instruments like the DX-7 have some of the most iconic presets out there, and they sound incredibly good on their own, and do not necessarily need any further tweaking.

How do you do it? Feel free to share your techniques and tips!

One technique I have been using lately is to take a sound, play it at c3, bounce it to paul stretch (plugin that can create really long smearing effects by stretching the sound), stretch it at a given time (lets say x200), bounce that back to Ableton (or your given DAW), and then load the stretched sound into a sampler and play a chord off of it.

Another is taking a sound and throwing it into a reverb with a really long decay time, then bounce that to audio and use just the tail part, which is mostly generated by the reverb at that point. It doesn't always work but it can get interesting results nonetheless.

So let's hear yours!

Exactly what Hytyma said, but I would the seemingly no brainer of try before you buy, nearly every company gives out month or so long trials, and they are a fantastic tool for deciding.
As for plugin recs, the ones listed are good but I would also add soundtoys to that list too.