Hi try this site for ideas
http://www.manchester2002-uk.com/info/tourist-information.html
I'm sorry but I don't know much about the links from the airport into Manchester but I would imagine they are pretty good since the recent Commonwealth games were held at Manchester.
There are also 3 good National Trust Properties fairly near the airport if you get a taxi ......if that is your kind of thing :
Quarry Bank Mill - at Styal approx 5miles. Particularly recommended if you are into industrial history . It is a beautifully restored textile mill & workers village.
Arley Hall near Altringham - Gardens & Hall - approx 10 miles.
Tatton Park - Gardens & Hall approx 15 miles
There is also Jodrell Bank Observatory approx 15 miles away for the space/ scientifically minded.
I would definitely recommend the Lowry centre and surrounding revitalised Salford Quays area. There is also a good science museum in Manchester too.