http://www.thetrainline.co.uk/

Hi there, train services between Manchester Airport and Manchester Piccadilly are run by two companies. Northern Rail run local stopping services and a few fast ones too and TransPennine Express run mostly non-stop services. All of them run to Manchester Piccadilly and Manchester Oxford Road but not Manchester Victoria. To get to Victoria, you will have to change onto the Metrolink Tram system at Piccadilly. The Tram Station is under the Main Station at Piccadilly and is well signposted. Its four stops on the tram from Piccadilly to Victoria and you want a tram with the destination BURY for Victoria. Both the trains and the trams run frequently, but if you want detailed information you can go to:
http://www.gmpte.com/info (For Trams) and
http://www.nationalrail.co.uk (For all Trains)

Go to the airport railway station (located off the walkway between terminals 1/3 and terminal 2). All trains except those going to Crewe will call at Piccadilly station – those operated by First Transpennine are quickest, while the ones operated by Northern Raill call at all the suburban stations on the way, but there’s only around 10 minutes’ difference in journey time.

The advice given in another reply about using the Metrolink trams to travel between Piccadilly and Victoria stations is normally OK, but does not apply between now and September, as there are NO trams operating in the city centre because the tram tracks are being replaced. The easiest way if you don’t want to walk between the two stations is to take the free Metroshuttle bus service – take Metroshuttle bus 1 from Piccadilly station approach to Deansgate, and change there to Metroshuttle 2 which’ll take you to Victoria (though it’s not very far to walk from Deansgate). Metroshuttles 1 & 2 operate every 5-10 minutes between 7 a.m. – 7 p.m. Monday to Saturday, every 10 minutes 10 a.m. – 6 p.m. on Sunday.