When to get in to Heathrow jump on the Heathrow Express which will take you to Paddington. I think you can buy a zone 1-6 "travelcard" from Heathrow which will allow you to travel as much as you want all day on London public transport.
London is actually quite easy to get round, a lot of the tourist attractions are in similar areas.
Kensington... ridiculously rich area that also includes the science museum, National History Musem, Harrods, Hyde Park. I'm pretty sure all the museums are still free entry with the occasional extra if you want to see a specific exhibition.
Waterloo... The London Eye, various thames boat tours, I-max cinema (ridiculously large 3d movies)
The centre... Buckingham Palace, Picadilly Circus, Covent Garden, Trafalgar Square, Bond Street. (Covent Garden is actually really good on a Saturday, there are loads of street performers there, highly entertaining)
I think those are the big three places, there are others but I wouldnt bother if you are only gonna be here for a day. (London Dungeons etc, but its a bit out of the way).
Personally if I was on a budget I would avoid Waterloo as all those things are gonna cost money. My own choice would be to to to the centre and take a look around there, Covent Garden is cheap entertainment (you arent obliged to give to the street performers, but you probably should give a little), you could take a walk round, go and walk through Green Park and Old Bond Street (possibly the most stupidly expensive set of shops you will ever see!), you can also see Trafalgar Square and Picadilly circus and there are a lot of tourist shops there if you fancy buying some souveniers.
Anyway, I'm rambling a little... what exactly is your budget, and what kind of things are you interested in?