From route 93, take exit 35 (A little north of Franconia Notch) or exit 40 if you're coming from the north. Your goal is to get on route 115 and take it to route 2, where you will turn right. Then on route 2 drive until you see a long parkinglot that parallels the road on the right. This is after Jefferson Notch road (both ends) and before Dolly Copp road. If you make it to Gorham, you have gone way too far. On a clear day, you will get nice views of the presidentials along Route 2. From Conway, take route 16 up till it meets 2 in Gorham, and then turn left onto 2, passing Dolly Copp road, and turn left into the Appalachia parkinglot.
From Appalachia you access the Airline Trail, one of the main trails up Mount Adams. You can also take the Shortline over to the King Ravine Trail, or the Valley Way over to Madison Hut. The Watson Path diverges from the Valley way, and goes straight up Mount Madison. Howker Ridge on Madison can also be accessed from this trailhead, via the Sylvan Way. The Link goes over to Lowes Path. Down low, numerous smaller paths connect the bigger ones. Most any trail on the route two side of Madison or Adams will lead one way or another to Appalachia. This is a large parking lot (photo at top shows most of it), however CharlesD says it becomes packed in the summer. This area does tend to be more remote and attract fewer people than Mount Washington.
Fortunately, unlike most areas in the White Mountains, you don't have to pay anything to access public land, since the forest service doesn't own the trailhead. It is plowed in the winter, though not as often as the main road. (see photo at top)
There are probably a dozen trails that converge on this trailhead, including Sylvan Way, Valley Way, Brookbank, Airline, The Link, Brookside, Inlook and Amphibrach. The Randolph Mountain club is to blame for the profusion. They publish their own map which you can find at http://www.randolphmountainclub.org/stuffwesell/trailmap.html ; most other maps (even the AMC maps) are not really complete.
Charles is quite right about the crowding; in summer it's common for cars to park on the shoulder outside the parking lot.