Welcome to the RS Helpline where fellow travelers offer advice.
Please take to the time to do the following. Using your computer's search engine type in Heathrow to London. When the available links appear click on tfl.gov.uk.
http://www.tfl.gov.uk/tube/stop/940GZZLUHRC/heathrow-terminals-1-2-3-underground-station?lineId=piccadilly.
At the top of the page click on MAPS. From options on the left side of the screen click on TUBE. Then depending on the size of your computer screen and eye sight either click on STANDARD TUBE or LARGE PRINT TUBE MAP. When you've accomplished this you'll be able to see that Heathrow is Zone 5, (located on the bottom left of the map.) Zones 1 and 2 are basically central London and the area where the majority of the "must see" sites first time visitors clamor to view. The Oyster Card will cover the nearly 50 minute tube ride from Heathrow into London.
http://www.heathrowairport.com/transport-and-directions/underground
You'll need to know where your hotel is located and the closest tube stop. Will it necessitate taking the Piccadilly line from Heathrow and then having to switch over to another tube line at Green Park to get to your final destination? That takes time as well. Will you be able to walk to your hotel or still need to take a taxi once you exit the Underground?
Yes, the Oyster Card can be used to get you to the Eurostar station.
Given your statement you dislike the idea of sending more money on travel in a city than the deal you got on your Eurostar tickets you'll probably disregard the following.
After a long flight from the West Coast of the States I don't wish to dawdle. So after a Passport Control welcome I prefer to get into London the quickest way possible. Thus I budget for a more costly mode of transport. The Heathrow Connect or Heathrow Express. Use a CC at a ticket kiosk and enjoy a quiet and comfortable 15 minute train ride into Paddington station. I'm then out the door, walking to my hotel. Solely my preference. Countless other posters wouldn't choose to spend the money but for me it's worth every penny. London is my favorite city on the planet and I spend as much time enjoying it as I can. Since the mode of transport to and from Heathrow has already been budgeted I don't give it another thought.
In over 12 visits to London I've never taken the hop on hop off bus. Very walkable city. Parliament, Westminster Abbey, The London Eye, Churchill War Rooms, Horse Guards, St James Park, Buckingham Palace, Trafalgar Square, The National Gallery and St. Martin's of The Field are all within walking distance of each other. Coupled with the ease of the tube and frequency of busses extremely easy city to explore. Buy the Oyster Card. Off you go.
Inexpensive is a relative term but for a safe, comfortable place to stay without costing an arm and a leg try the Vancouver Studio Apartments in the vibrant Bayswater neighborhood. Two tube stations, major bus routes along Bayswater Road. Can walk to it from Paddington Station. Easy walking distance from it into Kensington Gardens and over to Hyde Park. Well reviewed on Trip Advisor. Have stayed here more than once. Never dissatisfied.
Enjoy you trip to London.