Other downtown fun:

Head to P.E. Coles and Phillippe the Original for lunch - and taste the origins of the French Dip Sandwich

Asian restaurants galore:
Chinatown (and Monterey Park - great restaurants 20 mins. to the east)
Little Tokyo
Koreatown (just west)

Hip bars at night -
Golden Gopher and Broadway Bar, for example

Downtown is not so far from the east side... Silver Lake, Los Feliz, Griffith Park - even Hollywood - and as others said Dodger Stadium is right up the freeway - plenty to do.