Well it's quite vast as you can imagine and is split into a Dark Peak area and a White Peak area. Are you wanting 'run of the mill' touristy kinda places or somewhere a little off of the beaten track?
Castleton would be a good base for covering Winnats Pass, Mam Tor etc but it is busy pretty much all the time as it is easily accessible.
Dovedale is a nice place. There's a nice walk that follows the river and is very scenic. The tops there (Bunster Hill and Thorpe Cloud) are often less populated as they're pretty steep to get up.
Dovestone has 3 reservoirs, big car park toilet facilities and usually an ice cream van to boot. Plenty to explore around there and is not too far away from Manchester. There are big cliff faces which can give a great view of the valleys.
The Dove Valley is home to Chrome and Parkhouse Hill. Stunning views, can be quite quiet also. Close to Buxton.
The Roaches is another great place, only ever been once but it's a beautiful location. Plenty of free parking in a large lay by and is close to Macclesfield and Leek. There is also Hen Cloud which is over the road. The Roaches are very popular with rock climbers. Sheer drops, some of the biggest rock formations I have ever seen and a stunning tree top canopy to look over.
Obviously you also have the 'Dambuster' locations which were used to practice dropping bombs, Ladybower, Derwent and Howden Reservoirs, these all have a lot of surround high ground such as Win Hill (which eventually joins up with The Great Ridge), Bamford Edge and Whinstone Lee Tor.
The Chatsworth Estate could also be worth a look, plenty of walking and wildlife. Then there are the edges such as Curbar, Frogatt, White, etc etc.
You could pull in some waterfalls at Padley Gorge, Lumsdale, Three Shires, Monsal Dale too. Monsal Dale has a nice viaduct which you could photograph. Great at sunset as is Surprise View.
Hope that gives you a few places to look into visiting for the time being
@anibap - if you want a few more quieter spots then I'll come up with a list for you.