Which time of the year?
Pre-Covid I always used to recommend flying into Auckland and out of Christchurch or v.v to avoid back tracking. At the moment the only option is Auckland - and how long other flights take to come back is anyone's guess - it won't be clearer until much later in the year.
The main highlights are
Rotorua - Hobbiton
Wellington - Weta Cave and a lot of original locations but whether tours to do those start again - who knows
Queenstown - again lots of locations - many on private land
Fiordland
Remember we still have next to zero overseas tourists (only Australian's are allowed in) - so most of the more specialist / expensive tours are shutdown - maybe permanently. Hobbiton and Weta Cave are definitely operating .There are plenty of tours happening in many parts of the country - but the locals have kinda been there done that with LOTR .
If you can afford to private hire planes or helicopters I'm sure you'd be able to put together some bespoke tours out of Queenstown and maybe Te Anau - would be amazing, but not cheap.
On the other hand the country really does look like it does in the films - literally by hiring a car and say driving Christchurch - Wanaka via the Lindis Pass you'll see some pretty jaw-droppingly beautiful scenery that will look very familiar .
With 3 weeks I'd probably prioritize the south island