The fastest route to Angus and northeast Fife is the M90 motorway from Edinburgh. Exit onto the A90 at Perth (45 minutes), and travel an additional 20 minutes to reach Dundee; 15 minutes later, you'll arrive in St. Andrews. If you're coming from Fife, you can also take the A91 and the A914, and cross the Tay Bridge to reach Dundee. This is a slower route, but does take you through the heartland of Fife.
Fife and Angus cover a compact area, so getting around is straightforward. You can visit everything via a series of excursions off the main north–south artery, the A90/M90, which leads from Edinburgh to Aberdeen. From here you can easily make day trips to Edinburgh, Glasgow, or Perthshire. Aberdeenshire and the Central Highlands are not too far away either.
Fife is easy to navigate—although it can be difficult to find a place to park in St. Andrews. The most interesting sights are in the east, which is served by a network of cross-country roads. Angus is also an easy region to explore. It's serviced by a fast main road (the A90), a gentler road (the A92), and several rural roads that run between the Grampians and the A90.