Hiya,
I've lived on the outskirts of Manchester for a few years now and had to find out these kind of options when I moved here (although MK12 was my first Collectormania...so I've never been to this do at Gmex before) There is a train from the Airport which goes to Manchester Piccadilly.
From there you have a few options -
you could get a tram (routes to Altringham and Eccles all stop at Gmex and they run around every 10 minutes most of the day)
or you could get a mainline train to Deansgate station (which is pretty much where Gmex is)....routes to Liverpool, Barrow-in-Furness etc.
Or a taxi (there are plenty of them!)
Or even the 'free shopper bus' as I think that goes to Deansgate (via a very convoluted route round the one-way system)
However, depending on the amount of stuff you have with you to carry, and how much of a hurry you are in, you could walk it in 15-20 minutes, as the City Centre is pretty compact and no-where is very far away. If you want this option give me a shout and I will (attempt) to give some directions using shops etc. as landmarks.....or depending on the time you could follow me (as thats the way I'm going)
HTH a bit? Maybe?