There is a place called Bodega Wine Bar that has fairly cheap appetizers and glasses of wine for about $8 each. It is not the best wine bar, but it is cheap and worth checking out. I've only been the to Pasadena branch, but there is also one near West LA, in Santa Monica.
814 Broadway (@ Lincoln)
Santa Monica, CA 90401
If you want to try a real wine bar, not a restaurant with a big wine by the glass list, try the Gourmet Cheese and wine bar, in the southbay galleria in Redondo beach, it is 2nd floor of the mall next to Nordstroms. 20 white and 20 red wines by the glass, several interesting flights, incredible ambiance, Panini sandwiches, salads, meat and cheese plates, and unforgettable Olive bread. 1815 Hawthorne blvd Redondo Beach, 310-214-2122
To get an idea of the place try their website www.gourmetcheeseandwinebar.com