When I was in college, I was a founding member of the East Quad Congenial Bridge Club (EQCBC), the bridge club for our residence hall. Meeting each Wednesday evening, we'd form foursomes and play bridge, teaching each other how to play. At least one of the other regulars from EQCBC is now a regular player at bridge tournaments, so we must've done something right.
I'm also rather fond of Contract Gin. It's much like gin rummy, except it's played where each hand is a specific contract:
Contract Hands Description 6 33 Two 3-of-a-kind 7 34 A 3-of-a-kind and a run of 4 8 44 Two runs of 4 9 333 Three 3-of-a-kind 10 334 Two 3-of-a-kind and a run of 4 11 344 One 3-of-a-kind and 2 runs of 4 First 12 444 Three runs of 4 Second 12 3333 Four 3-of-a-kind 13 3334 Three 3-of-a-kind and a run of 4 14 3344 Two 3-of-a-kind and 2 runs of 4 15 3444 One 3-of-a-kind and 3 runs of 4 16 4444 Four runs of 4
Of course, each hand is still only ten cards (when played with the standard 3 or more people), so you have to "may I" multiple times in later rounds to get extra cards (the may-I card and a penalty card for going out of turn). It can be interesting, to say the least.