Air: Airports at Glasgow and Edinburgh offer plenty of connections both nationally and internationally. Additionally Scotair (0870 6060707) fly between London City Airport and Dundee.
Car hire is available from the major firms at both airports and in the city centres. See below for road travel information.From Edinburgh Airport, you can proceed by bus to centre of Edinburgh (Haymarket Station) and thence by rail to Leuchars. See below.

Rail: Nearest station is Leuchars (5 miles) on main line from London (King's Cross) - Edinburgh - Aberdeen. St Andrews may be reached by car, taxi or by service bus from Leuchars (2 to 3 buses per hour between 9am and 6pm and less frequently at other times). Travel by late overnight sleeper is convenient if met by car or taxi. Train time schedules available on-line:
http://www.nationalrail.co.uk/planmyjourney

Road: See sketch map below

(a) From south, cross Forth Road Bridge and proceed north along M90 to junction 2A. Follow the A92 until just before Cupar, then join the A91 to St Andrews. Alternatively, take M90 to junction 8, then A91 to St Andrews.

(b) From south-west, either go east along M8 to Forth Road Bridge and then by route (a), or by Kincardine Bridge and A977 towards Kinross, joining M90 at junction 6 and exiting at junction 8, then by A91 to St Andrews.

(c) From north-west leave Perth on M90 then by A913 to Cupar and St Andrews.

(d) From north via Dundee, cross Tay Road Bridge on A92 and after 1 mile via A919 to Leuchars and A91 to St Andrews.

Further detail can be seen at MapQuest here.
Also see the St Andrews map and index.