Last trip I took the underground from Heathrow to the St. Pancras. It cost all of ~5GBP each way. Flew on United into Terminal 4, took the elevator down 1 level from the information counter, and it was a short walk through an underground passage to the ticket station, and entrance to the train.
Below is a link to the official underground map
Checkin the Marriott map you can either get off at Westminister and walk across the bridge, or go past the hotel to Waterloo and walk back. Marriott website estimates both as 0.1 miles from the hotel.
If you buy a London Transport Oyster Card (combines the London Pass with a 7-day Travelcard) it is good for free export and admissions at many sites (lised of top ones below and how much you save by using the card).