In reply to cathalb55:
Unfortunately the area is NOT more gentle for Winter alpinism than the higher alps to the North, it can be very harsh at that time of year.
You will struggle greatly without a car - to the point where it is virtually unrealistic. In January, any car you may hire will require snow tires and chains. Often with the chains at least, it is cheaper to buy them at supermarkets than hire them from car hire companies for any period. Make sure you practice putting them on in relatively protected circumstances, i.e. don't make your first test of getting them on a live test!
Alpine climbing in the area, mostly in the Ecrins massif, in January is VERY serious! Particularly if you cannot ski-tour. Days will be short and at any sort of height (e.g. above 1000m), the cold may be brutal.
On the other hand, you could go snowshoeing in the Chartreuse, just outside the city, quite a lot of wooded paths to go along, mostly pretty safe from avalanches, but can be very confusing to navigate in. Some of them, like Cham Chaude or Charmont Somme are more straightforward to navigate in, also rather more open.
There is some Alpine like climbing in the Belledonne massif, but that is also more normally done rather later, toward April.
Most often, people will go there at that time for roadside ice-climbing. There is a little of this in the Chartreuse (if you pm me, I can tell you how to find it, but it is pretty flaky), more often it will be in the area of Bourg D'Oisan or further East. These are 1 - 1.5 hours drive out of Grenoble itself.
