Depends on what part of LA you are going to be around. LA is spread out. I would say if you are in or around Hollywood that Sunset is a must see. Grammen's Chinese theater is always great for checking out the celebrity foot prints. Grauman's Chinese Theatre - Hollywood - Home
www.chinesetheatres.com. Across from Grammen's is the Roosevelt Hotel that has been a Hollywood icon for decades. Celebrities from golden years like Clark Gable, Cary Grant hung out there as well as current stars like Will Ferrell--that is IF that sort of thing is what you like.
Down Sunset is the Guitar Center, which is a music store but in the street level they have a small museum with some music history stuff.
Olivera Street has early LA history, shops and Mexican Restaurants. Interesting walk area.
On 3rd and Fairfax there is the Farmer's Market with shops, eating places. It has been there since the 30s and attached to it now is an outdoor retail center called The Grove. Again, if you like seeing celebs, I have seen many there.
If you have the opportunity to venture further, would suggest Rodeo Dr in Beverly Hills or 3rd Street in Santa Monica. Venice Beach is also an interesting place to walk around.
Los Anglino Native.