Places to visit in Los Angeles

I usually end up doing odd but fun things a local Angelino does probably, well, only when someone is visiting (like go to Hollywood Blvd.) I will give you a list of a few must sees, and a couple of  day trip ideas.

Must-Sees:
The Getty Museums. There is one in Malibu (the villa) with all Greek/Roman art and the larger newer museum in Westwood. You need a reservation for the Villa, parking is $15, admission is free, and you get free parking  at the second Getty museum you go to if it is on the same day.
The beach (obviously) park for free by Patricks Roadhouse off the PCH and walk to through the tunnel for a good free beach day with good food and drinks right near by.
Wall Disney Concert Hall. Either catch Dudamel and the LA Phil, or look for a different type of concert if that isn't your speed. Pro-tip extra tickets are sold at discounted prices the day of (there often are none, so cal ahead) and there is a good student discount.
The Hollywood Bowl: tickets range from hundreds of dollars to just $10, and there are all different types of events. Pro-tip: pack a picnic (including blankets and trash bag) and feel free to bring your best wine or beer for most events!

Fun and Cheap:
Go to a show at the Upright Citizens Brigade for $5.  Pro -tip: get your tickets online and show up 90 minutes early to reserve your spot and grab a pitcher and a burger at Birds cafe, or get some coffee at the LA hipster Mecca: The Bourgeois Pig.
Check out the LA art walk in DTLA once a month or look for free shows at Levitts Pavilion (there is one DT and one in Pasadena... I'm more familiar with the amazing Pasadena location.)

The best resource for more fun events and places to check out is the LA Weekly Calendar (I'm not spamming, I am just an avid user) which has great search features, and an easy-to-use app.

Hope this is a good start, for more details or suggestions feel free to PM me, and welcome to Los Angeles!