Purchase the best travel insurance policy you can afford. Do not just purchase any old policy online, I would also talk directly to the insurers you choose re your travel insurance about your current state of health as well. Infact I would check this first before actually booking anything else, the travel insurance cost for yourself could well be very high indeed.
As you are planning to go to the US, an electronic travel authorisation called an ESTA will need to be completed for all of you in your party after you have booked the flights. This is done online (only use the official US government based website for this) and will cost $14 US (around £9 sterling) for each of you.
If your local airport is somewhere for instance like Manchester, I would try and get a direct flight from there rather than schlepp down to Heathrow.
Nowadays you can make the flight booking online and the process is quite straight forward. Once you've paid in full (which is what is expected with the airlines) you can at a later date print off the boarding passes. You can choose your seats in advance however, airlines like BA and Virgin charge you for doing so.
Luggage allowance varies between airlines but for economy passengers its usually one suitcase which weighs no more than 23kg. Hand luggage is separate and is usually one bag per person. Details would be on the airline's website anyway.
You can pack liquid items like shampoo etc in your suitcase, you cannot take that in your hand luggage or for that matter any liquid over 100ml through security. The liquids rule is still in force. You can buy water and other soft drinks airside to take on the aircraft.
Book a hotel for your accommodation needs in NYC. Short term apartment lettings under 30 days in that city are illegal so avoid sites that offer these. Also such sites do not verify that these listings are genuine.
I hope you do get to NYC and have a fab time.
Any other questions just ask.