So I'm still pretty new to the game and I can't quite figure out what the Hit and Critical Hit tokens are for. When a ship takes damage I've just been putting the cards from the damage deck by the Ship card. Have I been making a mistake?

Not sure which tokens you mean by "hit" tokens.

The red, hexagonal crit tokens are meant to be placed next to the miniature of a ship that has a face up damage card as a reminder to all players that there is so e sort of effect on that ship (reminding them that it is PS0, or can't shoot this round, or tajes damage when it overlaps or whatever). Basically it's a reminder. Not many people actually use them though (although you are supposed to,so in a tournament you should definitely have some).

What are the other tokens you mean?

There is hit tokens used in scenarios. Has 1 on one side and 2 on other. They are used for things that don't have cards, escort shuttle, satellite, etc