The Pennine Cycleway (National 68) and National 71 routes pass through Appleby.  

The Sea to Sea and Walney Wear routes pass near Appleby, at Penrith and Kirkby Stephen respectively.

Generally there are plenty of opportunities to go cycling along quite country roads.  The Exploring Eden leaftet (Anne - this is a broken link) includes four circular "do in a day" routes around Appleby.  There is also a "Cycling from Appleby and Kirkby Stephen" leaflet with two OS mapped routes drom Appleby and three from Kirby Stephen, which is available free of charge from the Tourist Information Centres in both towns.  

You can find lots of information on cycling on the Eden District Council website.