LOADS of suggestions! You are talking about one of my FAVOURITE places in the world!!
1st off...you HAVE to have Indian food while you are there...there are lots of vegetarian dishes & London has the BEST Indian food I've EVER eaten. There are even full vegetarian indian places where you can feel free to order ANYTHING.
Depending on where you are staying, there are lots of options...ask your concierge or a friend, they will DEFINITELY know! If you are near Piccadilly there is a WHOLE street of full of Indian places--near where I think "Grease" is playing.
My ABSOLUTE favourite things in London to do are:
-Go to the Tate Britain (not the Tate Modern, but the original one with all the OLD art).
-Go to the V&A (Victoria Albert museum) that seriously takes almost ALL day, be prepared!
-Go to the National Gallery & National Portrait gallery--I could sit in there ALLLLLL day. Seeing paintings you've only seen in books is amazing.
-Go to Lord Leighton's house (it's in Kennsington, I THINK) it's espectially great to see after you've been to the Tate Britain & seen his work--he's an amazing sculptor & painter.
-Go on London walking tours! Take the Jack the Ripper tour with Donald Rumbelow. Go to their website for info: http://www.walks.com
Also, I LOVED the places we visited on the Charles Dickens tour too. BEAUTIFUL places.
-Which leads me to going to Charles Dickens house. That's FAB! I loved that too.
-The Soames house museum
-MY FAVOURITE building & one of my fav places to go is the Natural History museum (it's next to the V&A.
-Of course the other touristy stuff like walking around parliament, big ben, westminster, etc, that's cool too!
Also, in the museums there usually are people sitting & painting the different amazing pieces to practice their art...you could totally sit in a museum & paint something of your own even just being inspired by one of the pieces. (I think you maybe just ask about it when you are there) And pics are usually allowed inside all the galleries, they'll tell you if they are not allowed.
Parks: I LOVE St James's park...it's one of the most divine places! They have Black Swans!
And Kennsington Gardens has a statue of Peter Pan, so cool!
And if you have time going to High Gate cemetary, that's great too! There are 2 sides of the cemetary, and BOTH close pretty early, so you should research it all before you go. One side you can only see on a tour, the other you are allowed to roam. The tour side is all we've made it to because of the timing issue (literally, a little old lade with a big key closes it down at like 4pm!)
Go to a pub & have a Shandy (my FAVOURITE!)==beer with lemonade (they call it lemonade but it's like 7up or sprite)...nice & refreshing!
OH--and there are no "restrooms" ask for the toilets. Hee hee! Otherwise, they get a perplexed look on their faces when you say "restroom". :)
Shows are expensive, BUT you can usually go up to the box office day of the show & see if there are any seats available & you might get a cheaper seat--and they have a TKTS & discount ticket places all over--especially at Leicester Square.
Get a GOOOOOD map--my ALL time fav is:
The London Mapguide by Michael Middleditch
It's the BEST...easy peasy to use, there are lots of things listed in there it's small to carry & tube info in there too.
OH! Also, depending on how long you are there...the 1st morning you are ready to go out & about & use the tube, get a week's pass it's cheaper. :)
And for a CHEAP easy sightseeing tour, grab a bus schedule, hop on a double decker & get up front & at the top. Ride around a bit & get your bearings. It's a little confusing to understand the layout of London from taking the Tube, although it's the BEST way to get around, still a bus is quite good too!
Have SOOOOOOOOOOOOOOOOO much fun!!