What needle size for what yarn?

Needle size depends on yarn weight - every ball of yarn has information on it about what needle size you need. That's the most important info, and you shouldn't ignore it, even if you think you could work with a different needle. The manufacturers have thought this through, because the right combination makes sure your project doesn't turn out too loose or too tight.

Generally speaking: thin yarn (fingering weight, 1-2mm needle size), medium yarn (worsted, 3-4mm), thick yarn (chunky, 5-6mm or more). But honestly, I wouldn't rely too much on rules of thumb - really check the label. Some yarns are surprisingly thin or thick compared to others in the same category. If you're unsure, you can also do a gauge swatch (roughly 10x10cm with the recommended needle size) and see how it feels.

A little bit of wiggle room is possible - some people deliberately go up a size for a looser finish. But for your first project, I'd stick with what the label says. After that, once you get a feel for it, you can experiment.