Auckland to Rotorua can be done in around 2.5 hours (via State Highway 27), faster depending on your exact starting point and route, and of course if you intend on travelling via car or if you drive slower in a campervan.
If you follow State Highway 1 and then SH5 it's estimated to take 2.5 hours so it's to you which route you want to take.
As for Picton to Chch following State Highway 1 should take you 3.5 hours.
Chch to Queenstown shoud take you 6 hours sticking to the main roads.
Don't tiki tour.. but if you want to, get a map so you don't get too lost.
Everything is well sign posted. Stay to the left. If you're going under 100kph on the open road pull over every now and again and let the traffic pass you. There are often passing lanes on the State Highways but drivers always appreciate the slow ones letting them get pass.
Source(s):
www.wises.co.nz is a good site for basic driving directions.
I travel Auckland to Rotorua often.