Was in London just last year. Had a blast.. I thought the city was so beautiful...
We aimed to do both touristy stuff and real everyday stuff to truly absorb the city:
Touristy stuff:
1. Visited most of the famous landmarks. Their city map pamphlet already cites all the popular places, complete with cute picture icons. It was fun to visit them and cross them off one by one. We did London National Gallery, Tower of London (they come with an option to get on a ferry ride that takes you back to central London for like 10 euros), The Parliament, The London Eye, and be sure to check out Millenium Bridge. It's a small footbridge that crosses the Thames river. It's just a bridge but it's so damn cool because it's small, narrow and accessible only on foot.
2. Hit the bars in Soho. It's a brick road neighborhood where it's littered by bars everywhere. We did some barhopping that was mighty fun.
3. Walk around, walk around, walk around. That whole square area next to London Eye and across from The Parliament is where all the bustlin' takes place: street artists, mimes and performers playing, etc. It's cool to just snap pictures. We stayed at a hostel and it's also a great place to meet other international tourists. Our hostel mates were this cute Aussy chick and another Aussy dude (no relations). Turns out, many Aussys work in England because pounds are 3x of AUD. It's these little stories you hear that makes the trip so great and intimate.