Glasgow Prestwick Airport
This is a tiny airport, previously more or less exclusively used for charter flights, it is now a fairly big base for Ryanair. Even with the recent refurbishment and expansion this airport is still tiny, and has very few facilities (after security there is a small duty free shop and tiny coffee shop and that's it). Make sure you buy lunch in Glasgow before you get here, because the prices are pretty steep.
Getting there by public transport
Prestwick is currently the only airport in Scotland with its own train station, and in fact this appears to be its main selling point. Trains run from Glasgow Central Station (40 minutes, every 30 minutes). The service is just part of the Glasgow suburban network, so they don't run late at night or early in the morning. Tickets are available for half-price from any station in Scotland to Prestwick on production of your ticket (see the ticket for details), which is obviously a pretty good deal.