Whatever you do, don’t eat at any national chains. Lots of good suggestions in this thread. I see tourists on Hollywood Blvd going into McDonald’s and I want to grab them and shake them and yell “YOU CAN GET THAT SLOP BACK IN IOWA!, HERE, TRY THIS!”
If you really need a burger, go to In-N-Out or Original Tommy’s or even Fatburger. Someone uphthread mentioned Father’s Office, which I have not been to but has a good rep.
Sorry for the partially duplicated post, the site is acting wonky.
Speaking of which, where does one see random celebrities?
We do intend to eat at In-n-Out. Dying to know what all the fuss is about.
I have been Jonesing for a Tommy’s chili burger ever since you posted that… but I’m 500 miles away. 
(don’t they call it a chili size?)
I’ll second the Huntington Library. In addition to the library (more like a museum- it has on display a Gutenberg Bible, an original of Audubon’s Birds of America, and many other amazing things), it has a huge garden where by just walking along a path you’ll pass through desert landscapes, a jungle, Australian plants and landscapes, Chinese and Japanese gardens, etc. There is also an art collection. Amazing place where you could spend an entire day.
In-n-Out is good and all, but if you’re going to eat there do it because you don’t have time to go somewhere else. Father’s Office, mentioned twice, has a pretty hard to beat, albeit much pricier (around $11 if memory serves) burger. More importantly, Father’s Office 2 is in Culver City, right off Venice Blvd. I still kind of prefer Umami burger, which has several locations but can be really hard to find, since they have very low-key signage. The Roscoe’s Chicken and Waffles in West LA also isn’t far from Culver City.
As a perpetual tourist in LA - I’ve been working there for the past 1.5+ years and will fly there again in a couple of hours - I was underwhelmed by the Santa Monica farmer’s market. The bluffs over looking the pier at Santa Monica are pretty cool. You can also rent a bike in Venice or Santa Monica and ride along a paved path that runs along the beach, assuming the parents are in fairly good shape and the weather is nice.
One place I really liked was Vasquez Rocks park, which is a LA county park. If you’ve ever seen a sci-fi or cowboy movie there’s a very, very good chance that you’ll recognize this place. It’s in almost everything, it seems. It’s free, and you can hike around or climb on the rocks. Only downside is that it’s pretty far from where you’ll be.
Another popular tourist thing is Olvera street in downtown, which is also not far from the famous Philippe’s one of the two places where french dip was supposedly invented. That in turn is not far from Chinatown, which is sort of a let down - all of the Chinese live in Monterrey Park/Alhambra/San Gabriel(?) now. Along those lines, I like to occasionally drive through Koreatown because it really feels like you’ve left the USA. There is good Korean food there, plus in Gardena. Good Japanese in Torrance - actually there’s a good Japanese market / food court on Centinela in Culver City called Mitsuwa Marketplace. They have a food court with good, inexpensive ramen. Good Thai in east Hollywood (Thai town) - Jitlada is off Sunset and is one of the rare places in the USA that focuses on Southern Thai (think lots of seafood and very spicy curries). Tito’s Tacos is a kind of an interesting throwback taco place, with tacos the way they were before authentic became the trend, but it is still packed with Mexican customers. This is in Culver City just south on Sepulveda from Venice Blvd. There’s another taco place in front, so go for the one you have to wait in line.
For random celebrity sightings, I heard The Ivy is a paparazzi goldmine.
I think the Santa Monica Farmer’s Market depends on the day you go as well. The Wednesday market is actually the largest and many of the big-name restaurants in the area will buy some of their produce from there. I’ve heard of celebrity sightings there as well, but I’ve never seen one the few times I made it there on Wednesday. Of course, I’m terrible at recognizing faces, so that might be part of it as well.
The Saturday and Sunday markets are both a little smaller, and the Sunday market is especially geared towards family-friendly experiences, so they’ll have more prepared foods but less vendors overall.
For food advice, you can never go wrong with Pulitzer Prize winner, Jonathan Gold’s reviews. His 99 Essential Restaurants in LAis a good starting point.
Or grab it just as you are leaving. There is one by LAX.
Most famously, where Kirk fought The Gorn in “Arena”.
Yes, it has fallen on hard times. Dirty, rats running around, and a lot of places with a “C” in the window. As ShibbOleth said, Koreatown is a much better choice.
Never got the appeal of this place. Very pedestrian tacos. It is closer to Culver City than Mariela’s, but not nearly as good.
My number one recommendation? See if you can get your flight changed and spend a few more days here. We have certainly given you enough to do! 
Is a major tourist trap. Whatever you do, don’t eat there. The establishments prey on out of towners that they know won’t be coming back. The places look inviting, but it is all decor. Expensive and bad!
I’d trust that opinion over mine. I haven’t spent any appreciable time there. It looked nice and fun on the outside.
It is a fun place to go, but most of the eateries are to be avoided. I also haven’t been in years, so my info may be out of date.
Well, it turned out that we ended up with even less time than we thought - we didn’t even make it to In and Out! We did get to take a city tour which was very good and let us at least briefly visit a lot of places, but I really wish we could have done more.
We did notice that our hotel was less than a mile from Holy Cross Cemetery, so we walked over there right before closing on Tuesday and my fiance got to see Bela Lugosi’s grave, which was a really big deal for him.
I can now admit that I was there to be on Jeopardy!, though, which is a relief. 
