Here's a link to bus services in Derbyshire. The 173 serves Monsal Head
http://www.derbysbus.info/index.htmI've often walked in the White Peak, accessing all the walks by public transport. I stayed in Buxton. Some of the most beautiful places to walk are the dales of the Dove - Dovedale and Wolfscote and Beresford Dales; the dales of the Wye - Chee Dale, Millers Dale, Water-cum-Jolly Dale and Monsal Dale; and Bradford and Lathkill Dales.
Derby has a good bus service to Ashbourne, the SW1, which would facilitate walks in the southern part of the White Peak, but if you wanted to walk in that part it would better to stay somewhere there, such as Ashbourne.
I wouldn't myself choose Derby as a base for walking the White Peak, as it's some way outside the area. Buxton and Bakewell are good bases and I think the travel logistics are easier than from Derby.