OK, firstly, relax you'll have a great time in London. Heathrow is not in London as I'm sure you've figured out so if you're staying near Heathrow then you're in Middlesex, not London.
Here are ways to get out of Hayes:
Tube
Although not in Hayes, Hayes' nearby tube station include Uxbridge the Uxbridge to Rayners Lane branch of the Piccadilly and Metropolitan lines and Ealing Broadway
Buses
Hayes has the following bus routes travelling through it: U3, U4, U5, U7, 427, 607, H98, 90, 195, E6, 140, 696, 697, 207 and 698. There are also several night bus routes.
Trains
Hayes and Harlington station offers frequent local services to London Paddington in about 15-20 minutes, and services to Oxford, Reading, Slough, and Heathrow Airport.
Road
The area is close to junctions 3 and 4 of the M4 Motorway. The A312 is the main north-south route.
www.nationalrail.co.uk (trains) or www.tfl.gov.uk (tube/bus) for times.
The London Eye is great but buy in advance otherwise you'll queue for ages. London bridge????? Are you sure you don't mean Tower Bridge? London Bridge is just boring and made of concrete. The Americans made that mistake years ago and paid for London Bridge thinking they were buying Tower Bridge. The previous one is now in Lake Havasu in Arizona. Trafalgar Square is nice if it's a good day. You can also go to the National Gallery while you're there - it's free and you can see Van Gogh's Sunflowers. Buy yourself an A-Z and you can't get lost in London because every street is in there and you're never far from a bus stop or tube station in the centre. Make sure you book a taxi to take you to Heathrow from Hayes. Use a registered minicab service (your hotel will probably recommend one).