First, my favorite place is the Musée de Grenoble. It's a fine arts museum that features works dating to the 15th Century all the way to contemporary art. Some of the works displayed include artists such as Picasso, Matisse, Kees Van Dongen, Modigliani, Gromaire and Andy Warhol. The museum also holds temporary exhibits from time to time. Although the museum is small, I really enjoy the atmosphere and the paintings. The size of the museum is not overwhelming like some can be in Europe, and most of it can be seen in a day. I enjoy going for an afternoon when it's empty and just walk around, sit, and enjoy art. Very calming!
Second, the mountains! Being as we are surrounded by them, they are hard to miss. Although I have not visited all of the three ranges, my favorite is by far Vercors. During one of my first weeks in Grenoble, I went snowshoeing there, and it was by far the most amazing experience. It was a beautiful, breathtaking day and the mountain was covered with snow. It was just so beautiful. The Chartreuse, however gives the beat view of the city. And a quick trip up to the Bastille, is a great workout!
Finally, my favorite thing to see and do is walk by the Isère river. It's really close to my house, and in a nice day there's nothing more relaxing than taking a stroll along the river.