Personally I would suggest using Sheffield as a base instead of Chesterfield, I am not sure about the buses from Chesterfield but certainly with the trains you will have to get a train to Sheffield and then change there for stops in the Hope valley, eg Edale, Bamford etc etc and due to connection times this will add time onto your day as well as being more limiting. There is a station "Dore & Totley" which is in a nice area of Sheffield although I am not sure about what holiday accommodation is like there. The other thing with Chesterfield is I personally feel it is a bit out of the way even if you are driving you will need to drive for a while before hitting the "action" so to speak whereas in Sheff it is very nearby, eg grindleford is about 15 min on the train.
I have just got back from 4 days in the peaks, I stayed in a self catering house in Buxton and used that as a base to go round where I wanted to go. Because I live in Nottingham the western part of the peaks is a little far for a day trip and so I used this as an opportunity to go on some long day hikes to the roaches, luds church, wilboarclough, shutlingsloe, macc forest, errwood, most of these places would be described as "wild moorland" type landscapes which is great for me and I will have explored some places that I've not been before. It depends what you want, there are a lot of woods and moorland near Grindleford, Edale has obviously got Kinder Scout which can get quite busy, to the south you get Mam tor and Winnats Pass, Chinley although more remote also offers good opportunity especially if you go to the south rather than the more explored north / Kinder area. Have stayed in the Lamb Inn near Chinley and definitely recommend it... Buxton is good option although a bit "end of the world" place in terms of public transport, so if you are arriving by rail from London you may be able to go to Manc and change there for the train to Buxton.
To find places I would like to explore I usually take an Ordnance Survey 1 : 25000 map and just look at it for a time - I know I like walking in remote terrain and woods / moorland / steep ascents descents etc so I look for that on the map. Once you get an eye for it you can plan a route fairly well. Figure out how the paths and roads join up and should be ok to create a route. I would say 90% of paths in the peak district are well marked / signposted it just tends to be on the fringes the paths tend to be a bit sketchy. If you have a GPS app on your phone and / or the OS maps app then so much the better.
Happy to provide more info about any specifics if I can, been walking around the peaks on and off for a few years.