Others have made great suggestions on how to get to your hotel. You really don't need the HEX!

Yes you can buy an Oyster card at Heathrow tube station but you can also order them online (TFL website as above) and have sent to your home so your ready to go as soon as you arrive.

Things to do? well depends what you like!

Personally I prefer more in-depth looks at places rather than a look from outside but on the top of my list would be

British Museum
National Portrait Gallery
National Gallery (both next to each other on Trafalgar Square )then perhaps a walk down Whitehall past Downing Street etc to Parliament / Westminster Abbey*
Greenwich - National Maritime Museum, Royal Observatory*, Painted Hall, Cutty Sark* Park etc.
Queen Elizabeth Olympic Park and the Orbit*
British Library

Apart from * the above are free except for special exhibitions

All the museums / galleries have excellent websites so you can plan you viewing in advance for specific things that you are interested in

You might want to consider a pass which if used properly may well save you some money on entrance fees if you are attracted to the things on them.