There are two good ways to get to Auckland. From personal experience they are pretty much the same. One Auckland route doesn't pass over the Kaimais. Take the Bethlehem bus from the centre of town to the Countdown supermarket. Cross the road and wait there. This route heads along SH1 towards Waihi, Paeroa, Ngatea. If you are heading to the Coromandel Peninsula just jump off at Waihi and go up from there. This is probably the easiest way towards Auckland.
The other way to Auckland is the road that heads over the Kaimai ranges to Matamata (Hobbiton). The best place to hitch is to take the bus to the Tauriko Caltex petrol station. Wait either at the station or at the other side of the road. An alternative spot which works well but isn't strictly legal is to wait by the vetinary clinic at the very start of this same road. It used to work well, but a no-stopping sign has been put up, so use with caution. There is a roundabout at the top of the hill called Barkes Corner. The road is marked towards Hamilton. To get there the easiest way is to probably take the Pyes Pa bus to Barkes Corner.
South to Rotorua, Taupo, Wellington
Again, there are two routes. The direct road (Pyes Pa) goes directly to Rotorua, and might be easier because of no-stopping signs being put up on the other road (SH2). Take the Pyes Pa bus as far as Joyce Road and hitch from there.
While it may still be possible to hitch on SH2 east, it may be risky because no-stopping signs have been put up. Find your way out to Baypark Speedway by taking the bus to Bayfair Shopping centre. Then head along the main road until you see the big stadium and the road splits into two, SH2 going east and SH29 curving back towards town. It should still be do-able by waiting opposite the stadium.
Or bus to Papamoa shopping centre on Domain Road, walk west along Domain Rd toward the toll road and hitch with a sign from just before the big roundabout.
Alternatively take a bus out to Te Puke, from where there should be absolutly no problem getting a lift further East. All traffic passes through the main road (Jellicoe St), so you should be fine if you wait on the outer edges of town. Soon after Te Puke the road splits and heads either East or South. It is worth turning down rides heading to whakatane if you are heading to Rotorua.