Help! Cheapest ways to travel for my trip to UK?

Yes, an Oyster card is best for ordinary travel in London. It does not go out as far as Windsor, but the train fares are fairly low and there's no advance discount. If you want an absolutely lowest possible price to Windsor, you can get the Green Line bus from Victoria Coach Station (street outside, not in the station itself) but that may have more traffic delay.

Depending on where exactly you need to go in Hounslow, either the Piccadilly Line tube or a local bus could be better to and from Heathrow. Both are covered by Oyster. To get from Hounslow to Bath on your first day, the best route is probably Piccadilly Line to Earls Court, District Line to Paddington, and then a train from Paddington. The problem is that cheap pre-booked train fares are limited to the specific train of your choice, so if your flight is delayed or your trip to the storage takes longer than expected, you could miss your train and have to buy a full-price fare. To get from Marble Arch to Hounslow on your final day, I would suggest a bus to Hyde Park Corner tube station (it's straight down Park Lane, which isn't usually very congested) then the Piccadilly Line train.

I don't understand why you need to go from Hounslow to Durham via Heathrow. Whether you need to or not, the best way would be to get the Piccadilly Line from either Heathrow or Hounslow to Kings Cross and train from there. I suspect that the very high prices are because you're being booked on the premium-fare rip-off Heathrow Express train from Heathrow into central London. Just buy a pre-booked fare from Kings Cross to Durham and use Oyster to get to Kings Cross.

Thanks Philip. The reason I wanted to go Hounslow > Heathrow > Durham was to send off someone flying home, then head to Durham. However given your advice I think we will change up our plans a little and say our goodbyes at Marble Arch.

Also, I've found a self-storage right next to Paddington Station. After looking through the London Tube Map, I'm thinking of taking the Piccadilly line from Heathrow to Hammersmith, then taking the Circle line to Paddington, leave the station to store my luggage, have dinner then take the train from Paddington to Bath. Does that sound viable to you?

Andrew, your last question in your follow up post, yes that seems reasonable to me.

If you have a credit/debit card with cheap or low fees that is set up for contactless payments, you might try using it on Transport for London tubes and buses. If it works - check their website for help - and many foreign ones do, you can tap in and out with that card. You don't have to tap the card on the reader which might give nefarious types a leg up, you can have it in a wallet or holder. Some phones also do NFC (Near Field Communication) which is what is used for both these cards and phones as well as the Oyster.

If it works you won't need an Oyster, you can register the card so that you can check on line what was charged, and it has the same caps both daily and weekly as an Oyster.

I'm lucky that I have UK cards which work wonderfully - my AMEX credit card and my bank debit card. They are so good that I got rid of all my Oysters last year.

There is no start up charge.

Do be sure of the charges your home bank puts on it before you try to use it.

See more about Britain's advance- and roundtrip-discount train tickets, which would bring down the high price of travel to Durham. If you have another long trip that might be done by train within the borders of England, a BritRail England pass (same page link), costs $221 for your 3 longest days of train travel within a one-month period (or $177 for youths under age 26) (does cover Heathrow Express if used on the Heathrow-Bath travel day).

You might find the King's Cross to Durham route cheaper if you split the journey at York. (I would certainly take a look around York).
The Paddington to Bath ticket could well be cheaper by splitting the journey at Swindon. Go on www.nationalrail.co.uk to find fares and click 'details' to find out when the trains are at York/Swindon. Advance purchase for cheapest fares for specific trains.

Thanks for the advice guys. I had a look at the Britrail website and it does seem convenient to get the Britrail England pass. Perhaps I can use it on the Paddington-Bath, Bath-Oxford, and Paddington-Durham days on the 3-day pass. I'm looking to do Oxford-London by megabus.

James, splitting the journey at York really would be quite a bit cheaper when I checked National Rail, thanks! It was like a 8-10 pound difference. There isn't much difference when trying to split Paddington-Bath at Swindon though.

Convenient, yes, to have a pass and just wave it. I don't believe it would necessarily be cheaper than well planned advance tickets, and I thought - based on your introduction and the title of the question - that cheapest was most important.

The best Paddington to Bath split may be at Didcot, rather than Swindon. Sometimes both ...

If you are happy using the bus from Oxford to London, also check this for London to Bath.