What do I do with a Foil trading card (Like a regular one, not the summer badge)
Foil cards are rarer versions of normal cards. You need to collect all the other foil versions in the set to craft the foil badge, which gives you additional experience for your steam level, and I think it can give you bonus rewards, like emotes and backgrounds?
So yeah, either collect them via trade, card drops, booster packs or the market.
Or if you're not bothered, which I'm usually not, then you can sell it on the market. They usually sell for more than their non-foil counterparts. Little bonus in your steam wallet.