Hanmer Springs is about 1 1/2 hours drive from Christchurch, 3 1/2 hours drive from the Picton Ferry and approx 1 1/2 hours from Kaikoura.
Driving from Christchurch includes some nice options for a tea break at Culverden (the Red Post), a beer at the Hurunui Tavern or a coffee break at Amberley. Coming from Picton you've got great options in Kekerengu and Kaikoura (incl Whale Watching).
There are several companies providing shuttle services from Christchurch though I'm not sure where they take off from these days (due to the Quake damaging shuttle offices).
We drove to Hanmer Springs from Christchurch...it took around two hours to get there and we got some lovely views of Canterbury along the way. It is an easy day trip from Christchurch, although be warned that you may be a little drained/tired after being in the thermal pools... ;)