Hi
Well London to Southampton (or Soton) is about 80 miles from Heathrow and about 60 miles from Gatwick there will be buses from both airports to Southampton for around £20 ($30?) or you could get the train quicker but lots more money about $50 I guess but you would have to travel in to London and then change to another train depot there are 7 or 8 big terminals in London, the coach bus is your best bet check out the bus companies website at www.nationalexpress.co.uk
I checked times takes about 4 hours and costs £22 single
Good luck, email me if you have any other questions about London etc..

Steve