Traveling by car is the best and easiest way to explore the area, especially if you get off the main, and often crowded, arterial roads and use the little back roads. The main route into both the Borders and Galloway from the south is the M6, which becomes the M74 at the border. You can then take the scenic and leisurely A7 northwestward through Hawick toward Edinburgh, or the A75 and other parallel routes westward into Dumfries, Galloway, and the former ferry ports of Stranraer (nearby Cairnryan is an active ferry port) and Portpatrick.
There are several other possible routes: starting from the east, the A1 brings you from the English city of Newcastle to the border in about an hour. Moving west, the A697, which leaves the A1 north of Morpeth (in England) and crosses the border at Coldstream, is a leisurely back-road option. The A68 is probably the most scenic route to Scotland: after climbing to Carter Bar, it reveals a view of the Borders hills and windy skies before dropping into the ancient town of Jedburgh.