Good on ya for getting on 2 wheels 
While the Summit Road and the Port Hills are nice road I'd advice not to head that way until you feel pretty confident on your bike, there are lots of nasties on the road - especially this time of year.
Go to Cashmere or Mt. Pleasant and get used to riding up-and downhill. Make sure you are comfortable stopping and starting both going up and down and that you are used to cornering both up-and downhill before even contemplating heading into the hills out of town.
When you feel comfortable with this I'd suggest going to Sumner, over Evans Pass, through Lyttleton and along the bays toward Diamond Harbour and then come back the same way - back to Chch either through the tunnel or back over Evans. If you are contemplating Dyers Pass you ought to ask Biggles2000 first to make sure it isn't covered in grit...
Have fun and be safe
It is preferential to refrain from the utilisation of grandiose verbiage in the circumstance that your intellectualisation can be expressed using comparatively simplistic lexicological entities. (...such as the word fuck.)
Remember your humanity, and forget the rest. - Joseph Rotblat
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