Dimmingsdale (not sure of a decent pub round there)
Ramblers Reteat cafe does great food and has a licence it’s not exactly the Peak but just as picturesque in an older woodlands, river valley kind of way
If you can find a route around Barlow, just north of Chesterfield the Tickled Trout pub is highly recommended (going there myself tomorrow) If no walks in the immediate area it’s only 10 mins drive (back towards Derby) from many footpaths around the various ‘edges’ of the east Peak
Closer to Derby grab an OS map and look around Cromford, Matlock, Ambergate, Crich area, plenty of walks to be had with some industrial heritage thrown in and plenty of decent pubs (ridings not bad either 😀 )
Hurt arms at Ambergate is ok again (used to be great , went poor for a while now new owners and decent Sunday lunch again) or The Dog Inn at Pentridge
This is quite a useful website – http://www.walkinginderbyshire.co.uk/walks.php
That draws together walks from a load of different websites, you will need to track some of them on an OS map first though as they don’t all have maps and directions are often interspersed with descriptions of the surroundings