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. 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.
There are local low-level trains from both Queen St and Central Station. The closest of these to Cairncross House is Exhibition Centre with many regular trains from Central Station. The walk is about half a mile Exhibition centre to Cairncross House [google walking directions].
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. 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.