A good way to plan your trip might be to get a copy of the NZ Lonely Planet book.
Being a long narrow country most people start their travels in the North (you'll probably fly into Auckland) and finish in the South, or vice versa. To see a reasonable part of both islands by road I would recommend at least 3 weeks. Some people take months. The following is roughly in order from North to South.
You'll need a vehicle if you want to see out of the way places. Many use campervans for a tour of NZ.
Roads are generally not motorways, but 2 lane roads with a 100km/h speed limit, and pass through lots of towns. This link will help figure out how long it takes to get places:http://www.aa.co.nz/travel/time-and-distance-calculator/
Although the North Island and South Island are of similar area, The South island has about 25% of the people and 75% of the scenery
Given your interests, I think you'll want to minimise the amount of time spent in the cities
Bay of Islands 3 hours drive North of Auckland are a popular tourist spot, with the main activity being scenic boat trips. The main other attraction is the history around colonisation. If you're trading off going here vs spending more time in the South Island, I'd pick the South.
For mountain biking, Rotorua has a great collection of purpose-built single tracks (best place in NZ) which are rideable in all weather because of the free-draining pumice soil.
Matamata advertises itself as Hobbiton, however all there is to see is the set for Bilbo's house. Most of the scenic mountain locations are elsewhere in the country - mainly in the South Island. Here's how to find the others:http://www.newzealand.com/nieuw-zeeland/feature/the-lord-of-the-rings-trilogy-filming-locations/?rd
For North Island hiking with LOTR significance you could do some of the walks at the link below. Mount Ngauruhoe is Mt Doom.http://www.doc.govt.nz/parks-and-recreation/places-to-go/central-north-island/places/tongariro-national-park/
The ferry across Cook Strait is worth doing during daytime to catch the scenery. About an hour of the 3 hour trip is through the Marlborough Sounds.
In the upper South Island, you can either travel to Kaikoura on the East Coast which is a popular whale-watching destination, or head west to Nelson. The Abel Tasman National Park offers hiking or sea-kayaking around the coast. For glaciers, you'll head from there down the West Coast to the Fox Glacier and Franz Josef Glacier. These glaciers go almost down to sea level. You can take a guided walk onto the ice - they provide crampons and ice-axe. If you want to do a helicopter ride anywhere, this is the place. The West Coast has high annual rainfall but is spectacular even when it's raining. However you could get a week with no rain. A hiking trip you can do from near Fox Glacier is about a 4 hour walk to the Douglas (? I think) hut which is by natural hot pools.
Near Nelson there is this fairly new cycle trail:http://nzcycletrail.com/trails/dun-mountain-trail/
From the glaciers you can head to Wanaka and Queenstown which are lakeside ski resorts - popular in all seasons. Queenstown has a multitude of tourist attractions.
There are various old railway routes around Central Otago which have been converted to cycleways - mostly gentle easy riding.http://nzcycletrail.com/explore-trails/
For hiking in the Fiordland area, check out some of these:http://www.doc.govt.nz/parks-and-recreation/places-to-go/fiordland/places/fiordland-national-park/
If you are interested in the Routeburn but it's booked out or you don't want to do the full length, a (fairly long) day trip can be to start from the Lake Howden end of the track, walk to the Harris Saddle and exit using the Deadman's track. You'll need to organise transport at either end. A cool place to base yourself for a couple of nights would be here:http://www.gunnscamp.org.nz/
If you're into feats of engineering, the (underground) Manapouri Power Station is great.
Given that you're probably pushed for time Invercargill and Dunedin could be given a miss unless you want to go to Stewart Island, and you would see more interesting landscapes returning to Christchurch via the Lindis Pass and Mount Cook.
The weather is best from January to March, with fewer crowds at the tourist destinations from mid Feb onward.
Unfortunately New Zealand has no trombone shopping destinations of note.
Good luck with your planning.