If you're planning to see a few shows I'd recommend The Woman in Black, Les Misérables and the Blue Man Group from what I've seen over the past year. There's some info on each and more here (CLICK!)
but this is just the 1st site I came across so I can't vouch for the 2for1 offers.
The Woman in Black was very good and a massive surprise as I'd read the book beforehand and still cannot see how it's so highly rated. The stage version fixes all the annoyances of the book and even openly calls it long-winded. Definitely worth taking a look, you'll be shocked at what this 2 (3!) man cast can do and the theatre alone is old and spooky.
Les Misérables I don't think I need to explain, simply amazing show and cast.
The Blue Man group I won't even try to explain, just go. Trust me, there is nothing quite like it. All I will say is that it's one of those you can pretty much guarantee that people of all ages will enjoy, just bring your sense of humour - And cross your fingers that you don't get picked to go up on stage...
Lastly try to get down to The Comedy Store (This is probably the least expensive if I remember correctly). The Comedy Store Players are on a few nights a week and do a show, all completely improvised, based on your suggestions. Very talented group of people, some of which you may recognise from TV.
Edited to agree with RNDdave, the galleries and Natural History Museum will be well worth taking a look at. I also forgot Chicago, another amazing show I'd definitely recommend.