Take a look at www.kayak.co.uk - input your airports, dates and adults/kids, it'll bring up all the flights available by all the airlines - you'll get the best deals too. It'll show you who is advertising those flights at those prices and provide the link to the site (such as expedia, lastminute.com or even the airline direct).
You'll probably find substantial cost differences between airlines such as Continental and Virgin - the former being cheaper.
Spend some time playing around with it and don't book the first time you search - quite often prices go up and down and research tends to lend itself to higher prices when booking at the weekend as more people doing so.
Book your hotels separately using the same site - it'll bring up best rates and again sites to book through.
Cost wise, it's going to vary widely depending on the rating of hotel and where abouts in the city you stay.
I've just gone through a similar process of searching/booking, so happy to be PMed if you want any help.
Tuesday tends to be the cheapest day of the week re flying.
Weekends as well have always been more expensive re flying (that also covers Friday as well).
DP should look at doing a fear of flying course; a lot of people have that fear and airlines like BA and I think Virgin do such courses. Its better by far to properly address it now rather than when he is actually on the aircraft.
I totally reccomend BA, with Virgin a close second. US carriers are cheaper, but the service is really bad. Plus, BA let's you book your family seats together, no extra charges, and two other points:1) you can look months in advance as to what the cheapest prices to fly any given day are (though they can change daily), and 2) the upgrade to premium economy usually isn't much, with better seats and double the luggage allowance.
Other tips: You can book BA through Boots treat street. Extra Boots points at no extra cost . Make sure you join the frequent flyer program. Two adults and two kids, economy and especially premium economy should be a free ticket to Europe. (Join something like e-rewards to get more frequent flyer miles).
Midweek definitely the cheapest time to fly.
Make sure of where you're staying in NYC. You don't want to be in a bad neighbourhood with young kids. We just got a very nice Marriott Courtyard for under £100 a night in midtown. There are cheaper (we booked last minute, but it can vary by neighbourhood drastically). There are Nectar points if you go through Expedia (through the nectar site).
Newark often cheaper to fly to than JFK and definitely not as insane. Easier connections (by bus) into the city IMO as well.