I'm very anti-Landing, but it's a good place if you aren't from St. Louis. It can definitely be a fun late-night area, but I don't recommend eating there. As for places to eat:
In Soulard, I like Llywelyn's Pub and Joanie's Pizza.
The best pizza in STL is either Pi in the loop (where Obama got pizza from) or Dewey's in University City or Kirkwood.
BBQ is Pappy's or Smokin Joe's
I have yet to eat at a bad place on the Hill and rarely go to the same place twice. Just pick a place there-you really can't go wrong.
My strongest recommendation regarding eating is to stay away from Washington Avenue (North of ScottTrade about 4-5 blocks). It is a major D-bag area and is a very over-priced area to eat in general. Schlafly tap-room is there, which is somewhat of an exception.
For breakfast, the Majestic in central west end (east of Forest Park) has great pancakes but is small. First Watch in Clayton is very good as well and bigger.
Things to do: The zoo is always nice during good weather. Forest Park in general is my favorite part of the city. City museum is cool (not a museum at all-more like a giant playground). There are 2 big major shopping malls: Galleria in Brentwood and West County Mall on 270 between 64 and 44. The arch isn't worth doing IMO. The loop is worth checking out, especially if it's a nice Saturday or Sunday afternoon. There are always a lot of people out there, and there are a lot of weird little stores.
I don't disagree with anyone's recommendations on here. My strongest recommendations are Dewey's and First Watch. Everyone go there at the same time please!