Miracle Mile
So Beverly Hills has been mentioned a few times in this thread so I thought I would give some more details of what to do in the area
From the 405 freeway you can get off on Wilshire and head into the city. You will pass many classic old stores like Neiman Marcus and Saks with their Vallets and very fancy items. The Buildings are beautiful.
You will continue and pass Rodeo drive, its on your left and you can take a drive right along it our get out and walk the street. One thing to know though is that some shops are by appointment only and will not let you in. That includes many of the jewelry stores, but you will still find many shops to check out if you want.
If you are a car person look around you will see some of those famous high end models and if you miss them on the street no worries. Up Wilshire is Lotus, Porsche, Lamborgini and other high end car dealers. Feel free to window shop, maybe you can have them wrap one up for you
As you head up further you will come to the Museum Row there you will find
LACMA-
http://en.wikipedia.org/wiki/Los_Angeles_County_Museum_of_Art
Its a modern art museum but often has great special displays like the King Tut one that was very popular.
Next door is
La Brea Tar Pits- This one is great to do with the kids. There is lots to see and do and a real tar pit!
Peterson Car Museum is great for the car lover with its interesting displays and more
California Craft & Folk Art Museum and Carole and Barry Kaye Museum of Miniatures are also there.
There is also theaters like the El Rey and Wiltern on Wilshire and lots of fun restaurants, clubs (like the stars go to) and more.
You can find parking structures with the museums or in other areas. Street parking can be tough here and you don't want to risk a ticket.