Glasgow's main airport (Glasgow International GLA) is about 7 miles from the University and has direct flights to/from the UK, Europe, the Middle East and North America (Continental have a direct Newark to Glasgow flight). Taxis and buses leave from directly outside the front of the airport terminal. A taxi from the airport is about £15-20. There is a fast direct bus (500) from the airport to city centre and a slower bus (747) that has several stops in the West End.
Glasgow Prestwick Airport (PIK) is further away and mostly served only by Ryan Air.
Glasgow's main train stations are Glasgow Queen St (trains to/from the North and East) and Glasgow Central (trains to/from South, Wales & England) stations. The main bus station is Buchanan St Station.
The Subway is the easiest way to get from the city centre to the West End/University (and its the third oldest underground system in the world). Hillhead is the closest to the University, though Kelvinbrige might be closer to your hotel (see map). In the city centre Buchanan St Station is convenient for Queens St Station and Buchanan Bus Station. St Enoch's is also useful for Central Station. Note that the subway is only open 10:00-18:00 on Sundays, but 06:30 to 23:30 the rest of the week.