"lots of cool restaurants and shopping"
I am most familiar with the section of South Fairfax from Beverly Blvd to Wilshire Blvd. There are tons of cool things in this area. The biggest attraction is the Farmers Market and Grove. I equally enjoyed strolling around the produce stand at the farmers market, then walking into the Grove and browsing some more upscale shops and restaurants. The Farmers Daughter Hotel on S Fairfax is very nice, very eclectic. I recommend staying here if you are going to CBS. CBS Television City is a very short walk across the street. There is a great Thai restaurant also right across from CBS that I highly recommend. It is not too far to walk from this area down to Wilshire and visit the La Brea Tar Pits. There seemed to be a lot of entertainment, food, and shopping on this street. It was fairly busy traffic-wise but not too congested or noisy for LA standards.