I traveled by train, bus and on foot, so not the same as you, but I don't have any hesitancy in saying that both of your proposed areas are lovely, and I think you'd enjoy either one a lot.
Annecy is particularly pretty, but its old town was the most tourist-thronged place I found during an 89-day trip to France last May-July. (I did not go to Paris or Mont-St-Michel.) If you choose Annecy, I suggest planning several options for out-of town activities, because I suspect you won't want to spend multiple full days in the old town. You can take walks around the lake (not always right along the shore), and I think bike rentals are surely available. A boat ride on the lake is very nice. I took a bus to the town of Talloires, often mentioned as the most picturesque town after Annecy; my verdict: attractive enough, but a bit underwhelming other than the lake view, and you already have that in Annecy. However, the late-June weather was very hot, which sort of elevated my standard concerning what was worth doing outdoors.
Wunderground.com confirms that I hit Annency during just about the hottest possible 3-day period in June (highs of 88F, 90F and 92F). If you have any timing flexibility, push your trip as early in the month as you can; that may help, weather-wise. If I had spent those same days in Dijon, I'd have faced 93F/95F/97F. Clearly, I was doomed.
Returning to the subject of day-trips from Annecy: Chamonix and Grenoble are possible, both about 2 hr. 15 min. one-way by train. I've read that the trip to Chamonix is especially scenic. I'm told that Grenoble itself isn't so great to look at but it has some good museums. Chamonix vs. Annecy is a subject that has been discussed more than once on this forum in the last two or three years; Grenoble rarely comes up. You might find earlier threads via the Search function. One thing is for sure: when the temperature reaches an uncomfortable level, altitude is one possible solution, so knowing what your options are would be a good idea. And be sure you have air-conditioned lodgings!
Edited to add: Bob and I were composing our responses at the same time. Like I said, the Annecy-vs.-Chamonix comparison is something that comes up fairly often.