Gatwick is about 30 min train ride to the city centre (London). It will probably take you to Victoria train station where you can get a taxi to your hotel. Est costs would be 10 to 20 pounds for the taxi and the train ticket I think is 15 to 20 pounds. You could take the underground (subway) from the train station but honestly I wouldn’t bother, the cost of the underground won’t save you that much and it’s more of a hassle. London taxi drivers know all the streets around so it’s never an issue of having them take you for joy rides.
Heathrow on the other hand is another story. It is about a 20-30 minute underground ride to the city centre. Taxi from Heathrow to your hotel is probably 50-60 pounds but I can’t remember the last time I did this. You could take metro into the city centre and if your hotel is close to the nearest metro and you don’t have too much to carry then that would be easier. I think Heathrow is the busiest airport in the world and I personally try avoiding it as much as possible. I am more comfortable flying into Gatwick, getting my bag and going through customs a little faster and getting on the train and then arrive in the city centre, after just get a taxi and you’re gold.