Don’t freak out - just google, google, google!
I live in London (well at least for another few weeks), so panic ye not, we can get you sorted!
Its extremely easy to get from Gatwick to London - just get the Gatwick Express
That will take you into Victoria, then if your hotel is near BP just get a black cab from Victoria to the hotel. I expect it’ll be about £10ish depending on traffic.
Expect to be jetlagged, and a bit of a space cadet for the first day or so.
Things to see and do - the parks of course (though they are a bit dry at the mo), do all the museums (science, natural history), I’d also check out what shows, and what exhibitions are due to be on when you’re here. Take a long walk by the river, and check out how pretty the city is at night.
London is a city that is best explored from ground level - it doesn’t really have the spectacular views of other cities (IMHO), but unfurling by foot it is extraordinary.
Most importantly - relax and take the time to have a pint in a pub! London is busybusybusybusy and everyone is in a huge rush to get where they are going. Ignore the rudeness and go with the flow…
When you take the underground - be prepared! Get your tube map and a travelcard (daily or weekly, zones 1&2 will be mostly what you need), take water (its a victorian infrastructure, so gets scorching hot), and pay attention to what is going on around you i.e. stand on the right of the escalators so busy people can runpast you on the left!
I also suggest googling ‘hidden’ ‘secret’ forgotten’ London and seeing what you come up with to look at - London is all about history.
Also (and most importantly) London is EXPENSIVE. Really really expensive. Take all the cash you can, and then take half again! Don’t eat in ‘Angus steak houses’ or other crappy tourist traps. That’s why people go home and say ‘the food in the UK is rubbish’ - well it is if you eat in places like that!!
Checkout square meal http://www.squaremeal.co.uk/ and http://www.thisislondon.com/ for good places to eat - we’ve got the most ethnically diverse cuisine in the world, and (don’t believe the hype about food in the UK being rubbish) some divine places to eat.
Good luck - enjoy! Have a google and feel free to ask any questions you want too.