score:4
In May this year I took a steam train from London Kings Cross to Matlock (peak district) and since then they have opened a 4 mile stretch and run regular steam train trips between Matlock and Rowsley (4 miles). Price wise it's currently £7.50 for adults and free for under 3.
Also the Peak District website looks like it has some good information.
Upvote:2
I can't think of any walks in the Peaks that are reasonably accessible by a push chair, I think a baby carrier is a much more reasonable option there. Also I haven't seen any spectacular things like waterfalls, etc. but I think there are some stone circles (not as spectacular as the Stonehenge though).
I did the Nine Ladies walk and it was very nice and not very strenuous (no push chairs though).
For rainy days (this year this will probably be your best option):
Upvote:2
Having spent quite a bit of time in the Peak District over the years, I wrote up a list of suggestions for friends. Many walks aren't suitable for push chairs, but you may find that the flat, smooth Monsal Trail (old railway line) works out. There is a nice cafe / bookshop at the old Hassop station on the trail. Various tunnels en route make the walk more interesting, as do the great views from the Monsalhead viaduct.
The path around Howden Reservoir, starting from Fairholmes on the Derwent arm of Ladybower, is also pretty flat.
The owls and otters at the Chestnut Centre (near Chapel en le Frith) sound like they would appeal.
The outdoor swimming pool in Hathersage is very kid friendly, and is warm enough that you don't need a perfectly sunny day for it to be nice.