To get from Heathrow to Belham, you can take the bus to Victoria Station for £5.70 cash. Comparatively, you can take the train from Heathrow to Paddington for £21.00. People wanting to save time will take the train, which is about 15 - 20 minutes; the bus takes a lot longer. It's a close call between the two because you'll be fatigued and lagged after a lengthy flight and clearing UK immigration.
For your larger question, you would be advised to get a Visitor Oyster Card for a one-time fee of £3.00. You top up the card as you go; the lowest amount to add is £10.00. They recommend getting a Visitor Oyster Card pre-loaded with £15.00, which should last you about 2 or 3 days if most of your travel will be in Zone 1.
Oyster Visitor Cards have a daily price cap on how much you will pay. The cap varies depending upon whether you travel at peak times or off-peak times. For example, if you travel during off-peak, the cap is £8.40, even if you take 20 trips. The price cap is also a feature on standard oyster cards.
You can always get a refund of the unused balance by taking the card to a tube station or sending it to them. They refund the balance in cash and return you the card with a zero balance.
You can order a card on-line and have it sent to you, it takes 8 to 14 days for an international order to arrive by post. Transport by Oyster is slightly cheaper and you will recover the £3.00 one-time fee in about 3 days of usage. Plus you can use it in future visits because Oyster cards never expire.
To order on-line, visit http://visitorshop.tfl.gov.uk/oystercard/product/oyster-card.html