the getty center in west LA is free. it's 5 dollars to park but there are shuttles and buses that go there as well. go at sunset, bring a picnic, check out the gardens and the view of LA.
along the same lines, LACMA and MOCA have a free evening once a week, i forget which but check their websites.
the galleries in chinatown and bergamot station in santa monica are good and once you get on their mailing list you can go booze it up at openings! bergamot cafe has nice lunches and you can eat dim sum if you go to chinatown.
bars: cha cha cha (two locations--one on virgil in silverlake) has an awesome happy hour with dollar beers and sangria and two dollar snacks. also cha cha lounge on glendale has a good happy hour (i think it's every night). the red lion, also on glendale, is another good cheap place to eat brats and drink good beer.
check out the los angeles board of chowhound for more cheap eating tips than you could ever work through.