There will be many open gardens and green spaces! Plus -
Also Worth Seeing
Brewhouse Yard - although currently closed due to the Castle restoration, from Peveril Drive you can still see the caves at the base of the rockface (see photo in the header above), and the interesting array of buildings, including the Museum of Nottinghamshire Life and Ye Olde Trip to Jerusalem - which claims to be England's oldest inn.
The Tunnel - built in the 1850's to provide a thoroughfare for carriages through to Derby Road, but never used. Folly?
Glimpses of the Castle - ducal palace that can be seen from various points throughout The Park, arguably the best view is from Park Drive