Take the Heathrow Connect into Paddington. (Not the Express – Connect is less than half the price and only 10 minutes more, leaving from the same platform. Buy tickets before boarding).
There is a Left Luggage place next to Platform 12 in the corner where the Connect train pulls in.
Paddington is very close to the parks and you can walk from there, or catch the Underground to elsewhere in the city. Suggestions if its raining include Bitish Museum, Tate, oh, any of the museums really.