In general, the size to use is related to what you are trying to do. For tiny surface mount stuff (like what's on the board behind them in my pic), you use small/tiny stuff. But for most things you'd do, the bigger stuff is just fine, I mostly use #5 size. It's big enough for pots and such, and if you're careful, good for even small things like resistors and such on circuit boards.
The main thing is how much heat it takes to get the braid up to temp. Bigger braid takes more heat, but also soaks up more solder. But too much heat can ruin small, delicate things like IC's and such. For general guitar and amp work, you're not messing whth too many tiny things, so bigger sizes are fine.
As for where to get it, I can't really say, I've got a supply I bought 5 or so years ago at an auction that a big electronics manufacturer was having when shutting down. But, I've bought a lot of things from places like Digikey and Newark, they are good to deal with. Heck, they have it on
Amazon...
I have NO idea what to do with those skinny stringed things... I'm just a bass player...