Listed from north to south...approximately:
Bleaklow: Appropriately named - it's very bleak up there.; it's the largest expanse of land over two thousand foot high in England. There are several dozen aircraft wrecks slowly sinking into the peat; it will be a test of your mapreading skills if you want to go looking for them...there aren't a lot of landmarks about.
Upper Derwent Valley: Consisting of the three reservoirs and dams; Derwent, Howden, and Ladybower. An easy walk around the shores of the reservoirs and spectacular views from high up on the moors.
Kinder Scout: The highest point in the Peak District; it's also a plateau like Bleaklow though and the actual summit is difficult to find...even if it's marked on your map. There are some interesting rock formations along the edges of the plateau.
Castleton: A tourist honeypot village, easily reached from Sheffield on the bus. Plenty of pubs and cafes and shops selling blue john jewellery, the local speciality...the mineral is still mined locally and is found nowhere else on Earth. There are four show caves in or near to the village - I think I've visited two or three of them...maybe even all four. The climb up to Mam Tor and then along The Great Ridge to Lose Hill is very popular and not too difficult.
Eyam: 'The Plague Village' - there's a lot to see for the visitor and it's a good base for walking.
Tideswell: Plenty of shops, pubs, and cafes. The parish church is known as 'The Cathedral of the Peak.'
Froggat Edge: The walk along the edge starting at Fox House and finishing at Baslow is one of my favourites.
Buxton: Grandiose Victorian spa town with regular cultural and artistic events throughout the year.
Monsal Head: One of the classic Peak District views.
Chatsworth House, gardens, and parkland: The largest, and one of the best, stately homes in the country.
Bakewell: The unofficial capital of the Peak District. Shops, pubs, a lovely riverside walk and my favourite reason for visiting...the lovely courtyards bedecked with flowers in the summer. Of course you can also buy a Bakewell Pudding from one of the three rival establishments each claiming to have the original recipe. You can even buy Bakewell tarts...if you really want to!
Monyash and Lathkill Dale: A pretty village and possibly the best of the limestone dales.
Lumsdale: A hidden gem. A series of waterfalls tumble down the hillside in this steep gorge where several ruined mills are located. It's only a couple of miles from Matlock.
Matlock Bath: Known as 'Derbyshire-by-the-Sea' due to its large number of amusement arcades and fish and chips shop...and other tourist attractions. Maybe you might want to take the cable car up to the Heights of Abraham or visit on a Sunday [especially bank holidays] to view the impressive array of motorbikes that will be parked along the promenade.
I'm not able to get over to the western areas of the Peak District...and I know there are a lot of nice areas and attractions over there as well.