For LA it's really hard to explain what to do and where to go since the city/metro is so huge, and it really depends on what you like to see. I take a shot at it and see if perhaps this may help you.
There are so many things I like to do, and they are all over the metro, and I suspect your time is limited in LA. Truthfully there are many tourist spots in LA that are in my opinion a must see for a first time visit. Yes even Venice Beach, even though most locals I know never go there unless they are taking a visitor.
Personally I would recommend visiting downtown LA (all of it and have some good walking shoes), Old Town Pasadena, Downtown Glendale(Brand Blvd and The Americana), and of course The Grove/Farmers Market in the Fairfax District of LA. Yes this is where the tourist goes, but these are local spots as well. One thing I will say about is The Grove and The Americana are designed by the same architect so they are very similar in look, although the one in Glendale is somewhat less touristy and blends in better with the downtown area of Glendale so I like it much better. But if you make it to the Fairfax District, visit the jewish deli Canters http://www.cantersdeli.com/
. They are open 24hours and have a very good pastrami.
Take a tour of the Red Line from Union Station (tour the station) to North Hollywood and check out the neighborhood. I would also suggest taking the Gold Line to Old Town Pasadena and check out that area, and also ride it in the other direction to East LA, no need to check out the neighborhood but really just to see this part of the city by rail.
If you like Museums? I suggest checking out the Norton Simmon in Pasadena, and the LA County Museum of Art. I was thinking it's too bad you're not visiting in the summer because they have a great concerts for free usually consisting of jazz on Fridays, and the latin jazz series is outdoors on Saturdays afternoon, and what a party that is, you can picnic on the grass and even bring you own liquor. I would still check out their schedule online at LACMA to see what's going on during your visit. Also Sundays Live at 6pm is free every week with classical/chamber music all year and most of the time it is quite good depending on your taste. Colburn School is another place you can listen to classical music for free or very cheap in downtown LA near the Disney Concert Hall.
For cheap Mexican but very good check out Ramona's. It's not in a tourist area, but over in Gardena on Crenshaw near El Camino College. There is no where to sit inside (well there is but only for employees) so you have to either take it home, or as many of us do eat in our cars in the parking lot of the strip-mall. The food is very good trust me I've been going there for many years, check it out on YELP. This is one place that locals definitely go to.
You may also want to drive up to Palos Verdes and check out the views of the LA Basin and the coastline from the peninsula. The scenery is breath taking and after seeing the views you will understand why people are willing to pay the steep prices to live there.
Other suggestions is taking a drive down Pacific Coast Highway, but starting out by taking Ocean Avenue from downtown Long Beach through 2nd Street Belmont Shores-Long Beach and head south on PCH and drive all the down to Laguna Beach. Trust me it is one of the most beautiful coastal drives of any urban area in the country. Of course it's not like taking Highway 1 down from San Francisco since there is more traffic and development since this part of the LA/OC metro area. After arriving in Laguna Beach I would walk around the downtown area as it's very walkable and check out the art galleries, restaurants, shops and take in the cool vibe to this area along with the beach as it's front yard. Another suggestion while you there would be to visit the Wyland Galleries. He is the guy who is famous for painting underwater sea life on many buildings around the USA. No need to spend a lot of time there, but it's worth just stopping in and perhaps even just sit on their patio facing the ocean and get a cup of coffee and take in the view.
I'll stop here since I could write a book on suggestions. Perhaps others can chime in and offer some other things to see. You may want to PM Kingofthehill since he's an LA guy who gets around the city and knows it quite well. Another person who I know lives on the west side of LA to contact via PM and knows that side very well is Illithid Dude. I doubt either one reads this section very often, so I highly suggest you PM them for their input, I hopes this helps.