The Pennine Cycleway (National 68) and National 71 routes pass through Appleby.
The Sea to Sea (C2C) and Walney Wear routes pass near Appleby, at Penrith and Kirkby Stephen respectively.
Generally there are plenty of opportunities to go cycling along quiet country roads. The Exploring Eden leaftet 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.
Information on Mountain Biking in the area can be found here
Please check out the Inspiring Cycling Website for information on cycling in the Eden Valley here.
You can find lots of information on cycling on the visit eden website.