If you want a nice walk by the beach, without getting sandy, head to Palisades Park where you'll get great beach views among palm trees and greenery. Also close to the famous Santa Monica Pier.
Great area for walking around, shopping, and eating.
Beverly Hills is great for window shopping, and if you go north of the famous Beverly Hills sign, you'll get to see some amazing mansions.
The Getty museum is known for its architecture and grounds, and is a nice way to feel cultured :)
If you have time to go east, the Huntington library and gardens is a beautiful place to spend an afternoon.
A few restaurant recommendations for while you're in LA: - Pizzana (Brentwood) - Laurel Grill (Culver City) - Zaytina (Culver City) - Hatchet Hall (Mar Vista) - Fig (Santa Monica) - R+D Kitchen (Santa Monica) - Milo + Olive (Santa Monica) - Rosaline (West Hollywood) - Zinque (multiple)