Some pretty inaccurate stuff been said so far.
I lived in GNB last year. It is the best ski city in the world bar NONE. PM me with specific questions if you like I can help you out with a bunch of stuff.
Closest resorts are Chamrousse and Les 7 Laux.
Chamrousse (25 mins from downtown) is ok but quite low and not very steep. Some decent touring to be done though from the top and you can get single-ride lift passes to the top for like 5 euros, then tour from the top.
Les 7 Laux (30 mins from dt) is fucking sick. Really superb lift accessed terrain and awesome touring options. Really good park too. Student season pass is 280 Euros.
Some little resorts in the Chartreuse range and Vercors range are fun for the odd day here and there. Tiny but fun. St Pierre de la Chartreuse can have great tree skiing.
Alpe d'Huez and Les 2 Alpes are the nearest "big" resorts (both ~1 hour). But any resort in the alps is gonna seem huge compared to the east coast. Les 2 Alpes has an awesome park on the glacier through the summer and autumn and a good winter park too. Its more expensive though and the freeride terrain is not great.
LA GRAVE (~1hour) is really second only to Chamonix in the world in terms of ridiclously gnar terrain and HUGE vertical. Going with locals is crucial though as much of the terrain is glaciated and/or massively exposed.
Any other resort in France is within 2 hours of Grenoble. Vars, Serre Chevalier, Paradiski area, Espace Killy area, Portes de Soleil area, Chamonix area all well worth making trips to. Parts of Italy and Switzerland within easy reach too. Grenoble is not called the "star of the alps" for nothing.
Most importantly make friends with local shredders to fully enjoy the best ski city in the world. Having a car really helps.
Like I said PM me for more info