Hi Sarah, all 5 of Sardinia's main climbing areas (Iglesiente/SW, Sassari, Cala Gonone/Dorgali, Isili and Ogliastra/SE coast) offer "single-pitch on-sighting and 1-day 'projects'? 6s would be nice, 7s required, 8s not necessary."
Asking for both limestone and granite narrows it down massively: only Ogliastra/SE coast offers both, and there are only three granite crags but they are really good:
1. Cava Usai (Villasimius, faces E, good for autumn/winter, lovely setting above the sea) See photo
http://www.sardiniaclimb.com/default/AlbumCopertine/pages/CopAndrea2.html
2. not as many routes but 3 superb 7a+/7b flared offwidths which would meet your onsighting goals at Lucertole al Sole in Lotzorai, we live 2 km away. Look up Lotzorai on it.wikipedia.org to see Johny Dawes onsighting one of them
3. porphyry: Praidas at Villagrande, remote setting, recent visitors felt they were in the film Deliverance.
All three are much better than Cala Spinosa at Capo Testa whose rusty bolts and 3 hours' driving distance from any other crags makes it not worh considering IMHO.
The only guidebook describing these crags is the 5th edition of Pietra di Luna, the guide to the whole island with 4000 singlepitch routes. If you base yourself on the east coast, somewhere between Cardedu and Santa Maria Navarrese, you have the 800 single-pitch routes of Ogliastra, Villasimius 1:30 away for a day trip and the Ogliastra crags 10-40 mins away. All angles, from the slabs of Serra Oseli and Genna Croce (Brits would call them walls) to wall climbing at Jerzu and the steeper routes on tufas at Ulassai (eg Cave of Dreams
http://www.sardiniaclimb.com/default/AlbumCopertine/pages/caveCop.html this is an 8a, very good 6c+'s to 7b+'s and a soft 7c nearby) and Baunei (eg Su Telargiu Oro, this is 8a/+ but there are great 7's).
The east coast also has some great MP routes, not all waiting for James Pearson and Carlone Ciavaldini to repeat them as in this "really makes you want-to climb here" trailer
Good luck Peter