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Some good tips so far.
I also recommend heading to the Southbank/Royal Festival Hall. There's always something going on there and much of the space is open to the public during the day to just hang out.
London also has amazing cinemas if you're looking for something to do on a rainy day. Very central ones aren't cheap, but I recommend the new Picturehouse Central cinema, which shows a lot of art house/non-blockbuster films. Then there's the Ritzy in Brixton (check out the market for cheap around the corner) and the Electric Cinema in Notting Hill (a super comfy boutique cinema).
Live music is also amazing. Check out what's happening on songkick whilst you're there. Places like Camden and Shoreditch have nice small venues with music every day.
Check out the parks, St James, Hyde, Regent's... all beautiful in the Autumn. Hampstead Heath too.
Apps like Flixter (cinema listings), yplan (stuff happening) and sites like songkick and yelp.
Tinder is also very popular in London, should you want to try and find a date/someone to hang with.