Oh goodness! You can't make a 3-week trip to California and not visit Catalina Island for a day (at least), if not overnight. The Island is located less than 26 miles off the coast and it is just gorgeous. The ferry ride across the ocean takes about one hour. One of the best things to do on Catalina is rent a golf cart and drive around, taking in the wonderful panoramic views and scenery. You can take tours to the inside of the island to see buffalo. You can take a boat out to what they call Seal Rock. If you are there during May - September, you can take a tour to see flying fish (my favorite). There is also a great semi-sub tour where you can see all kinds of undersea life, including sharks here and there. You can hike, rent a bike, rent a kayak, rent a pedal boat. Catalina is alternately fun and relaxing. I have done a few one-day trips there, and I even stayed there for a full 8 days once. I love it - it's my 'other' favorite place in Southern California (besides DLR, of course).
Start out by ordering the free Visitors Guide online (it is like a magazine):
http://www.catalinachamber.com/catalina/vacation-planner
And also, if you check out the first page of my TR (link to TR below, in signature), there is a sort of Table of Contents, and you will see links to two pages in my TR with lots of Catalina photos. You must visit there if you are going to be in California for 3 weeks!!
The Catalina Express (the ferry that takes you over to the Island) is right next to the Queen Mary in Long Beach. Also, the Aquarium of the Pacific is right there as well. That's worth seeing.
Also, I would suggest LACMA (Los Angeles County Museum of Art) and perhaps the Original Farmers Market (near CBS, so lots of celebrities go there to eat lunch). The La Brea Tar Pits are right next to LACMA, so that's good too. Sunset Blvd., Melrose, Santa Monica Blvd. and Hollywood Blvd. are all a short jaunt up Fairfax, which very close to LACMA and the Tar Pits. Beverly Hills/Rodeo Drive is just down Wilshire Blvd from LACMA, so everything is very close together.