By train... flying into Heathrow it's the Railair coach service to Woking, then a train from there. Flying into Gatwick you have to go into London and back out again (change at Clapham Junction). STN and LTN are entirely the wrong side of London. From LCY you take the DLR and Tube to Waterloo then a train from there.
Unless flights to SOU are really extortionately expensive I doubt you'll be saving much by flying to a London airport.
There is a direct National Express coach from Heathrow to Southampton, and a direct train from Gatwick. These cost £25ish. But take close to 2 hours. The train from Southampton airport to Southampton central takes 5 minutes (and the train station is about 30steps from the airport). So think about whether it is worth paying for the convenience.
My apologies for the incorrect information in my earlier post - there are indeed direct trains from Gatwick to Southampton. I was looking for today (well, "yesterday" now in my part of the world), and there weren't any more for today when I was looking a few hours ago. Should have picked a random day in the future and looked for daytime trains.
I have just been to amadeus.net to look up fares to SOU for the OP's intended travel dates.
Fares from Belfast start at GBP 55 for a one-stop flight and GBP 74 non-stop (all on Flybe)
Fares from Dublin start at GBP 34 for a non-stop flight
This is all on Flybe, though it would seem that one needs to book through either Ebookers or Bravofly to get the cheapest fares (Amadeus has the links)
The Belfast fares don't seem at all extortionate to me, and I doubt it would be cheaper to fly to/from another airport and take the train or coach.
Thats not going to work for someone looking to save money though, which is the entire point of the exercise :-)
The OP should just book direct with an airline (and certainly not with Bravofy as mentioned above) and I'd have said that with the possible exception of gatwick, which would still add about 3 hours extra travel time its hard to see it being worthwhile for a money saving reason, especially if you can get fares as low as suggested in post 7, eg around £50-£75.
And why should one not book with Bravofly if they are offering Flybe flights for less than Flybe themselves?
I was looking at amadeus.net (which is a price comparison site) and that's the information it came up with. Amadeus is usually pretty good at finding the supplier with the cheapest fare - sometimes it's the airline itself, sometimes it's a third party site.
If you are aware of any problems with Bravofly then do let us know - it would be useful information to have. But without concrete information I tend to trust amadeus.net, they have always managed to find me the cheapest fares so far.