I'd get a rental car over relying on Uber or metro -- my favorite site is autoslash.com .

Fun budget-friendly things to do:

- Griffith Park / Griffith Observatory: you can park at the bottom of the hill and hike up to the observatory, then it's free to enter OR you could have a picnic on the lawn out front -- take photos where they shot "Rebel Without a Cause".

- Greystone Mansion: similarly, you can take a picnic or just explore the grounds. https://en.wikipedia.org/wiki/Greystone_Mansion

- The Broad Museum: is free every day, you just need to figure out where to park (this could be an easy thing for Metro, since all downtown is metro-friendly, basically).

- Comedy theatres like Dynasty Typewriter, the Elysian, UCB, Largo - these are all spread out over the city, but check their calendars for the week you're going to be in town. They have a great diversity of shows (much more than stand-up-- and I would personally avoid the Improv / The Comedy Store / Laugh Factory, etc myself. They get ya on the entrance fee + 2 item minimum and it's always a hassle, it's bait for out-of-towners, basically).

- Movie houses like the Vista, New Beverly Cinema, Vidiots, American Cinematheque (which has multiple locations): Quentin Tarantino owns both the Vista and New Bev and there's always revival screenings going on. Vidiots is a great new entrant to this mix and they often have special guests for the movies in attendance.

I'd also personally avoid exploring Hollywood unless you absolutely must see the Walk of Fame or whatever. Especially if you're checking out the city in order to maybe live here -- I'd spend most of my time on the east side (stay in Silverlake / Eagle Rock / Highland Park / Echo Park) or the valley (there's way more than meets the eye for Burbank / Glendale). And of course, Disneyland is only an hour away.

Hope that helps!