Most or all of the airport have easy access to the city via public transport. Heathrow is on the Tube but Gatwick and Luton both have regular train/bus links too. You can sometimes even buy the tickets at a discount when booking flights. One you're in the city it's very easy to get around by bus, Tube, taxi etc. I'm not sure what travel cards are available these days but my mother was over there recently and just bought an Oyster card which can be topped up as you go and this worked fine for her. I think there's a refundable deposit on the initial purchase (e.g. £10 for the card with £3 or £4 as a refundable deposit and the balance in prepaid credit or something like that). Pick a few "must sees" rather than trying to get around the whole city as it's quite big with lots of sights/attractions.