Advice depends on what you really want.
If you just want to bag the peaks the Airline-Valley Way loop is the easiest way. It is also the least interesting, but since you will get spectacular views from both summits, and from the Airline, it is still a good trip.
Much more interesting is ascending Madison by the Valley Way -> Brookside -> Watson Path. Then ascend Adams by the Star Lake Trail. Many interesting options for the descent, I would recommend Lowe's Path to Thunderstorm Junction, then Spur Trail, Randolph Path, Amphibrach (a wonderful trail for tired feet at the end of a long day) and the Link.
The Howker Ridge Trail starts not far from Appalachia, and Howker Ridge to summit, down as described above except that you take the Randolph Path all the way to the Randolph East trailhead (where the Howker Ridge Trail starts).