Srinagar, Gulmarg and Pahalgam. Gulmarg will be nice to stay in midst of snow. You can plan your stay at srinagar for rest of the days and visit Pahalgam. Shared jeeps are also there.
All of the state but unfortunately you can't. So any place that doesn't have military restrictions is good in Kashmir, particularly from Srinagar onward. Once in Kashmir, make sure to visit the local tourist office that will advise you all about safe places to stay. I'd like to add that any place you go in the interior, and it becomes more beautiful exponentially, the more you go higher up or beyond Srinagar, if you can manage homestay, there is nothing like that for the people are really beautiful and lifestyle in remote areas of the valleys there is simply amazing and full of a promise of a rich local culture experience.
That question will not do justice to J&K. As you should probably be taking a month's leave to explore that place on earth. Every destination you visit in that place will hold you tight. You shall leave you part in those valleys once you come back.
Jammu and Kashmir, the Paradise of the Earth said the Mughal Emperor Jahangir, nobody can reject that this spot is honored with dreamlike ecstatic excellence. It is the Northern most province of India, imparting boundaries to China, Tibet and Pakistan, is likewise known for the most picturesque territories of India. Off-beat places to visit in india : 1) Pulwama
2) Kokernag
3) Tulian Lake
4) Yusmarg
5) Hemis
6) Lolab Valley