I want to be able to synthesize really unique drum sounds and effects (think Flume or Ta-Ku) but I have no idea how or where to start. I know how to synthesize a basic kick from a sine wave but it sounds very plain and generic. I'm not sure how I would even go about claps, snares, shakers, hats, etc. Or do people like flume just sample drums from other songs and run them through a bunch of fx? I would rather not use pre-made drum hits from sample packs as I understand you are very limited in the sounds you can create.

You'll make your life harder trying to make all your drum sounds by synthesizing them from scratch, but nothing is preventing you from layering different drum sounds combined with a bit of synthesis for certain frequency areas of your sound. Using well manipulated and/or layered samples to create one final drum sample will certainly not make your final sound less original or unique in the end, if that's what's your worried about.
If you want to use synthesis for learning purposes, fair enough, it's cool and great fun (though it can be quite frustrating when you find yourself failing at achieving the result you wanted) but I think it's important to realize where it's limit are, in terms where it can or can't (easily) help you achieve your final result (Hope this sentence is understandable). But if I could only work with synthesis for drum sounds, i'd go with FM Synthesis as it's much more flexible in terms of timbre than what a basic subtractive synthesizer could give you. Plus FM is much more convenient for bringing some inharmonic frequencies into your sound, which is often required for synthesizing drum sounds, but it's also more complex than other types of synthesis.

I'll leave that here. Maybe it will help you understanding the concept of FM synthesis for drums. https://www.youtube.com/watch?v=T5lgDDBdQyk

Also, good EQing and and bit of saturation/distortion in post processing could make quite a big difference in the final sound.