Los Angeles is a big city with lots of traffic. I live in Westwood near UCLA and Century City. It can take me an hour at times to get to Downtown Los Angeles (15 minutes with no traffic), Hollywood as much as 45 minutes (20 minutes on surface streets with no traffic), 45 minutes to Torrance (South Bay - 20 minutes without traffic), an hour to Long Beach (30 minutes with no traffic).
Most of the Los Angeles County tourist stuff is west of Downtown Los Angeles (Beverly Hills, Santa Monica, Hollywood, beaches, museums etc.). Pick a hotel from Downtown Los Angeles and west and forget the San Fernando Valley (Burbank). Just too much traffic to get to/from there at peak times (even weekends). I actually have friends who rarely travel from the SFV to the basin because of the traffic. I won't go at certain times to the SFV because of the traffic, either.
Century City is great & central but if cost is a factor, I'd pick the LAX area. Close to the beaches and not all that far to Beverly Hills, Hollywood (to the north and east) or Long Beach to the south. Traffic on the 405 in the morning northbound and in the afternoon & early evening southbound can be a real b*tch, though but one way or another, you will run into traffic.