Lucky you!
Chamonix is an obvious destination, easy as a day trip from Annecy (or several day trips). Aiguille du Midi is the star, but I love the view from Brévent. Mer de Glace... Well, the pics I've seen from the past few years are downright scary in terms of glacial retreat.
Closer to Annecy, the area around Grand Bornand and La Clusaz is very scenic. There are some good easier hikes at Col des Aravis (to Croix de Fer) with Mont Blanc views, also up on Plateau de Beauregard (Access by gondola from la Clusaz, or drive up Col de la Croix Fry). Plenty of chalets serve food, so a picnic isn't even needed (a few snacks is never a bad idea). And the forest south of Plateau de Beauregard has tons of blueberries and some raspberries by late July/early August. Do wash before eating blueberries though.
The Résistance memorial on Plateau des Glieres is very moving, and the area is very pretty too for, again, easy hikes. Great mountain food "chez Constance" up there (and I'm sure the other restaurants around are equally good).
South of Annecy, on a clear day, you can see a good chunk of the French Alps from Semnoz (Cret de Châtillon). Late PM is best.
There is of course Lake Annecy itself, with plenty to do: biking on the dedicated path on the western shore, some nice beaches on the eastern shore (I like Angon), a castle in Menthon, views at Roc de Chère. Right above the lake, the panoramic views at Col de La Forclaz are jaw-dropping. Great for paraglider spotting, too.
North, this time, you have Salève, a lowish mountain with some great views over Geneva and towards Mont Blanc, and a pleasant ridge drive.
Further north, on lake Geneva, Yvoire is super cute and definitely worth a visit. Some good restaurants, too (Pre de la Cure, Jardins du Léman), but make sure to book.
If the weather isn't as good (but not pouring), or if it just rained, Gorges du Fier is a fun outing. Walkways in a narrow gorge.
I could go on, I guess... There's really a lot around there. My grandparents lived 15 miles east of Annecy, and their house is still in the family, so I know the area quite well.
