There are about 10 ski resorts in the area, from Hanmer Springs up to North to Mt Dobson down South.
Roads are not the most developed thing in NZ some plan some time to get from one place to the other!!
We highly recommend that you have your own vehicle or a rental vehicle. There are might have some buses from Christchurch but they miht be expensive and the journey longer and less comfortable. Chains are seldom required, however they always have to be carried. Chains can be hired from Christchurch.