I was just there as a tourist last month and I can tell you a few things I enjoyed
-British Museum. It's gigantic so you should try to explore as many exhibits as possible, or you could spend a full day there and try and see everything. It's free too and has cheap food for sale.
-Tate Modern. If you're into modern art this is the spot. It has like 5 galleries but it doesn't take ages to explore it all. They also have an awesome lawn on the shore of the Thames where you can chill and have a coffee. Free as well.
-The Globe. I would definitely recommend seeing a play. I'm not the biggest theater fan, but this place was really enjoyable. You can choose standing room only tickets, which are really cheap at 5 pounds, or seated tickets which are a bit more expensive.
-Pub crawl. These are super fun. I went on one that was called the rock and roll tour and I got to see all the places in London that are important to rock and roll history. I also went to two pubs. As an 18 year-old American, this was the highlight of the trip for me :)
-Parks. I spent a whole afternoon walking through Hyde Park and eating ice cream and enjoying the weather. It's basically London's central park, so don't expect to walk across it quickly.
-You should also buy an Oyster card, basically a transit smart card that you can use on the Tube and buses. It saves you some money and is more convenient than buying tickets every time.
Hope these tips give you some ideas.