Where would you be working?
There's an apartment complex on Figuroa that we looked pretty hard at. It's one of the few places downtown that actually has grass surrounding it. Staples Center, if you live downtown, it cakewalk to get to. I drive past it everyday (it about 4 blocks over and ten blocks up) and its my halfway point to school. It takes me about ten minutes to get to teh Staples Center from the Bank District.
If living downtown isn't scary for you, there are tons and tons of options. Most lofts that have enough sq. footage to house a married couple go for 2,000-2,400 depending on location. Ours is 1,800 and about 750 sq. ft. There are a lot of two bedroom apts. and bigger lofts available too.
Alos, most people will tell you to go through WestSideRentals.com to find your place but if you're looking for downtown, save the cash and just drive around or check out theloftguys.com They've got a pretty good database. But really, just driving around downtown, you hardly ever pass a block without spaces for rent.
Expensive: 7th and flower to 3rd and grand can be a bit pricey. Everything youfind on Fig between 1st and the Staples Center is very expensive. Old bank District is pretty decent. Fashion District is too. all that area is cool.
We're in the pelofts.com Pacfic Electric building and its one of the better dog freidly places. Lofts go for anywhere between 1,400 (600ish sq ft. studio) and 3,500 (two bedroom, or two story penthouse with a view) a month. There's a dog run on the roof and a pool, etc. Secured parking. You've got about 700 neighbors in the building right now, too.
If downtown isn't for you, or if you're looking to rent a house, Pasadena is a good area, as is Altadina (we found a two br flipped house for rent for 2k or sale at 350,000 which is pretty good in LA).
Silverlake and Echo Park (where Doger Stadium is) are relatively hip areas and hous a bunch of college kids, as does Westlake (UCLA). Anaheim is an option, but is a sizeable trafficcy commute.
And there's some apartments to be found at the beach - Marina Del RAy, Long Beach (which can be a bit gang ish), Redondo Beach, or SantaMonica if you have a trust fund.
SF Valley isn't terrible, bu the commute can be to downtown during rush hour. But there are subway stops from Studio City to Hollyweird that can kill some of that commute. Simi Valley is almost out of the question. Its expensive and far.
Hope some of that helps.