any recomendations for biking new zealand north island?

I am presuming that your finishing point, with your friend, is also back in Coromandel. Yes? In that case, I also presume that from the East Cape , you will continue through Rotorua, tauranga and the East Caost beaches ( mostly native forest along the road) back to the Coromandel peninsula.

A couple of other choices you could decided on , are which way you go around the central plateau on the way down - you could go down the desert road side to Waiuoru, and coming bcak you could go the National Park viallage side, to get to Tongariro.

Or, do a wider loop. From Coromandel, head over to Hamilton and down through te Awamutu, Waitomeo, Taumaranui, wanganui, joining back onto HW 1 at Bulls. OR continuing on to Palmerston NOrth, then finding the back route to Levin ( it actually has cycle signs on part of it)

OR:Hamilton to Taupo on HW1.
OR: Hamilton to the dam at Atiamuri. Turn right there, and go through Whakamaru and the back way to Taumaranui.

The most direct route would be to go through Paeroa, Te Awamutu, Matamata, joining HW 1 between Cambridge and Putaruru ( ie: no Hamilton)

There is nice scenery both ways. Driving in NZ, I tend to stick to HW 1 because it is more direct. However, if I was cycling and had the time, I would definitely go the less direct route ( less traffic)

going through Taumaranui. ONe of the most spectacular pieces of the North Island for me, is the route from Turangi to Taumaranui. I also loved the Route via Atiamuri and Whakamaru, the one time I did it, 20 years ago.

BY the way, although this is a cycle site, if you are interested in the outdoors generally, ( and you must be, to be considering walking the Tongariro track) then I really recommend kayaking or canoeing the bush section of the Wanganui, from Taumaranui to jeruselum. You can hire boats in taumaranui and arrange to leave your bikes there, and be driven up again after the trip ( BLazing Paddles are one Company I recommend.)