If you're going to be on Steam, and become a gamer, the other suggestions over here are good.

However, if games aren't really your thing, and you just want a casual gaming experience, then I'd suggest a few Linux native games, that exist in all distro repos: gweled, ltris, lbreakout2, lgeneral, frozen-bubble, gnome-mahjongg, gnome-tetravex, xye, kobo-deluxe, aisleriot, powermanga, open-invaders, supertux, pingus, berusky, opentyrian (requires data from the dos game, which are also free to download elsewhere).

Then there are some more heavy hitters (still native linux games), like freeciv-gtk3, opencol, 0ad, tuxracer, lincity-ng, simutrans etc.