Westwood Village, which is right next to the UCLA campus, has some small stores. But within 5 miles, there are 5 major shopping areas.
The closest and biggest is 3rd Street Promenade in Santa Monica, which has all the major stores from Forever 21 to Club Monaco, a MAC store, a Sephora store, a Fresh store, and a Lush store. If you don’t drive, take the #1 or #2 Big Blue Bus and it’ll take you right to 3rd Street.
Down Westwood Blvd, there’s a somewhat smaller mall, Westside Pavillions. A H&M has recently opened there, and a Forever 21 is also opening soon. That mall doesn’t have the best stores, and but there is a Bare Escentuals store as well as a Ziba salon, which offers the best eyebrow threading ever. Posh Spice was reported to have shopped at Westside Pavillions before. Take the #8 or #12 Big Blue Bus and it’s about 10 minutes away.
On Santa Monica Blvd in Century City is Century Mall. This is the place with tons of high end stores like Louis Vuitton. There’s no direct bus route, but I believe you can take the #8 or #12, then transfer to the #7 on Olympic Blvd.
Then of course, just about 2-3 miles down Wilshire Blvd lies Beverly Hills, with Rodeo Drive and all its shops. There is traffic in Beverly Hills, but there are great shops there.
Just a bit further is Beverly Center, which does sit on Beverly Blvd, but isn’t actually in Beverly Hills. Tons of stores there!