Oh I love answering questions like these being a Londoner who has a true appreciation for the city she lives in :)
Well of course you should get a travel card, I think Transport For London sell ones that last that long? While you are here you will find this site extremely useful www.tfl.gov.uk
Use the journey planner, it will get you from A to B to Z and back to A again without fail. (and you can look up travel card details)
You should definitely see a play or three in the West End, absolutely wonderful.
You yourself mentioned Regent’s Park, I strongly recommend this! It is situated in the North Of London in St. John’s Wood which is a very high profile area with many celebrities residing there. Also home to Abbey Road Studios, where I think the Beatles/Paul McCartney recorded their songs.
In Regent’s park from May there is an open air theatre, you should look up the times/showings for productions there, I saw a production of Peter Pan it was truly wonderful.
You should visit places like the Horniman Museum in Forest Hill and of course the Tate, Tate modern in central London near the Millennium Bridge (which itself is quite a sight).
If you want to get away from the city centre more, more suburban areas that are good to go are places like East Dulwich, where there is Lordship lane, it is full of boutiques and shops you can go and visit, a very peaceful safe environment, as well as lots of greenery and parks. It is just really homely and full of interesting architecture, not to mention Dulwich Park. (I should know, I live there!)
Also you should visit Crystal Palace Park, home to the Crystal Palace Tower, it is simply beautiful. Look it up there are loads of things to do within the park.
Obviously you should see a concert or something at the 02 Arena, things like that will soon become obvious.
Portobello Market, Covent Garden, Camden and Borough Market are a must.
Depending on where in particular you are staying you will find your own little places you’ll like to visit. :) Have fun!
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Also across the road from Dulwich Park is the Dulwich Picture Gallery which you might prefer as it is far less touristy and more intimate place to visit. I am not sure about prices.
Also there is a place called Belair Park where ‘Carter’s Steam Fair’ visits around the time you will be here, I love going to it when it comes around!
You could also visit Brockwell Lido if the weather is good.
dfkldj sorry you got me going now!
hope i helped :)