I haven't begun anything of the sort yet, any idea how I should first approach this? Should I just pick a game, try and go as fast as I can until I stop dying and actually go pretty quickly, and then start optimizing my paths? Or should I watch vids first, and try to copy tricks right away?

get the basics and understanding down by watching vods for games that you may be interested in running to kinda understand whats going on. then maybe pick a few games you may enjoy playing and trying to play-through normally a bit at your pace then eventually start getting better and faster.

All right, thanks a lot, i'm still so indecisive on which game to start. My current selection has been narrowed down (but is not limited to) to Donkey Kong Country 3, Super Mario RPG 7 Stars, Paper Mario (64 or GameC) and Sonic Adventure Battle 2. I just don't know which one I want to tackle, I do know I want to try both any% and glitchless, I'll probably tackle 100% much later on.

Take a game you love. The speedrunning thing will be easier if you really enjoy your game, you'll spend hours without even noticing.

If you didn't find a game you REALLY love, you can take a game you think is interesting for speedrun (need good execution or has a lot of glitches for example). Being really good on a hard game or succesfully doing impressives glitches is something I personally can't get bored of. Just find what make you say "I want to speedrun!".

If you have absolutely no idea of which game you could speedrun then take a not too hard game with a good community just to get the basics of speedrunning. People are really happy to see new heads, go on their stream (if they have one, or just send them a message) discuss with them, share your experiences, your tips, etc... It's a good step in the speedrun's world, and you can always find a game you like to speedrun in the way. ;)