I wrote the following for a friend that’s planning on visiting London for the first time. Maybe some of you London natives or frequent visitors have more to add in the comments?
Running: I highly recommend running at Hyde Park/Kensington Gardens. Lots of running paths, beautiful views, fresh air, the Serpentine, beautiful gardens, swans and ducks at Round Pond, Kensington Palace to see where Prince William and Kate live, the Princess Diana monument, Speaker’s Corner, Marble Arch, and so much more to see as you run!!!
Touristy spots: I’d suggest taking one day and doing one of the double decker tour bus routes. You can hop on and off as much as you want and knock out all of the tourist spots like Big Ben, London Tower, London Eye, Millenium Bridge, The Monument, Buckingham Palace, etc.
Shopping: Every Saturday there is a great outdoor/indoor at Portobello Road. It stretches for about two miles and has lots of vintage, one of a kind finds as well as interesting and off-beat touristy shopping, too. Don’t miss this! There’s a place called Cockney’s Pie and Mash on Portobello Road that serves the most amazing pie and mash and eel, a local Cockney favorite. I HIGHLY recommend this. There is limited selection as they specialize in one thing: pie and mash. They’re homemade pastry covered meat pies with mashed potatoes in a parsley sauce. Optionally, you can get eel added into the sauce, which I also highly recommend even though it sounds totally bizarre. Delish.
Food: As noted, the pie and mash and eel above at Cockney’s. Fish and chips take away in a paper cone! Mushy peas are a British side dish that has to be tried (add lots of salt!). Get Indian food at a local place. It will be delicious and more authentic than the Indian food you’re used to in the States. Whatever you do, don’t eat at any tourist spots or chain restaurants. Use Yelp and read the local’s reviews. Find places off the beaten path. Contrary to popular belief, British food is delicious.
Misc: The rest of the time should be spent exploring local haunts, museums, galleries and the like. You can use Time Out London to find out what’s on. There are plenty of cool walking tours to try out, for example the Jack the Ripper tour was pretty interesting and ended at the pub he used to haunt for drinks!
Most of all, ENJOY your trip!