Actually from Buxton rather than a drive away… some quite pleasant mountain biking, there’s a classic route that goes over / round Ladder Hill, drops into Taxal (Whaley Bridge fringes) then back above Errwood Reservoir and over the old road back into Buxton.
You can also walk up that way and loop around above Errwood through the Goyt Valley. It’s all pleasant, though not mega techy. There’s ace road riding too on mostly back roads. The big hill above Dove Holes is a cracking walk/run and just along the A6, you could walk from Buxton I think, but I’ve never done it.
Buxton itself makes for a pleasant urban meander – Pavilion Gardens and the Opera House all quite scenic.
The obvious sub-20 mile route with more tech if you do want to drive to start would probably be the Jacob’s Ladder / Roych Clough loop out of Hayfield – or anywhere else on the loop I guess. It’s the one that V-Publishing released for free as an all-weather route, but is just a very good, very Peak, ride. The V-Publishing guide book(s) are good if you want to make a return trip some time – download link:
http://peakdistrictmtb.org/wp-content/uploads/2017/01/Peak-District-MTB-All-Weather-Route-Update-2020.pdf