A New Zealand mostly Lord of the Rings tour - three weeks

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

We did a three week self driving tour of NZ. We used a company called Fine Tours New Zealand. They arranged the car rental, accommodation, and some of the tours for us. It was very convenient and allowed us to not worry about the logistics of the trip.

We are not Lord of the Rings fans, but they do a whole tour based on that theme.

A couple of suggestions. We stayed in a variety of accommodation, but I would recommend staying in four star or better places. Also, explore more areas as NZ is a wonderful country to visit. Very friendly and helpful.

Have a great trip.

"With 3 weeks I'd probably prioritize the south island "
Me too.
There are LOTR film locations dotted all over the country, but one of the greatest concentrations of the spectacular scenery you see in the films is the area in and around Queenstown/Te Anau.
Doing your own planning and having a car will give you the freedom to explore at your leisure while booking the occasional LOTR day tour from Queenstown ... of which there are many.
There are a number of local guidebooks that specialize in LOTR-themed self-driving tours such as the one you propose. Picking up one of those will help you sort out your planning and craft the most efficient itinerary that works for you. Can probably find one on Amazon.
If you want to splurge on one thing I'd suggest one of the helicopter tours - wouldn't be cheap but it would be an experience like no other, and would allow you to visit some of the remote filming locations that would be difficult or impossible to do on your own.
A few logistical suggestions: book a car with automatic transmission versus a manual, which like Europe tends to be the default choice. Even if you have experience driving on the left having an automatic will make it a much more pleasant driving experience for you.
NZ has become a popular destination for visitors from mainland China, Taiwan and Singapore and gets particularly crowded during the Chinese New Year period that generally runs from around mid January into early March. Both accommodations and rental cars tend to book up quickly (and early) during that time so do plan ahead.
The popular and scenic drive from Te Anau to Milford Sound can be a dangerous one for the inattentive. It's a relatively narrow, two-lane road that's shared with tour buses and other large vehicles. Wandering out of your lane while mesmerized by the scenery can have fatal consequences - it happens several times per year.
And maybe one slightly off-the-wall caution: there are a number of scenic pullouts along the road down to Milford sound - great places for photo ops and short hikes, but the resident Keas can make a mess of a rental car while you're parked. They like to chew on anything rubber - seals, tires, windshield wipers, etc. Can not only be messy but expensive too if not covered by your insurance. Some companies have started offering car covers to help discourage their attentions. It's worth picking one up if you're headed that way. Ask me how I know about this ...

I did a self-directed tour. Will say right off the bat the sites are very widely scattered across both islands. You will need a book to find most sites.

North Island

The highlight is probably Hobbiton about an hour west of Rotorua. Note that the site preserves the set for the Hobbit films as they were filmed last.

There are sites all around Mt. Ruapehu in Tongariro National Park. The graphically altered Mt. Doom (Mt. Ngauruhoe) is also in the park, climbing it now discouraged if not forbidden.

In Wellington is the Weta Cave studio. Also in Wellington along Alexandra Rd is the site of many of the forest trail scenes (the Black Riders) the first days journeying out of the Shire. A long day trip from Wellington are the Putangirua Pinnacles.

South Island

This site doesn't get mentioned much but it really is the prettiest filming location https://www.atlasobscura.com/places/mount-sunday north of Geraldine in Canterbury, remote but good gravel road there. Makes for a very satisfying visit.

I realize there are a lot of tours from Queenstown but not too many sites near there. There's the 12 mile delta area just west of town, easy. There are other sites near Te Anau SW of Queenstown a 2-3 hour drive one way.

With 3 weeks you can see both islands, you have to in order to visit LOTR sites. Make sure you book your interisland ferry (they usually sell out and the pricing is dynamic, it goes up over time) and a one way car rental. South Island to North often has no one way rental fee. The fee can be a few hundred dollars the other way. From Illinois I recommend using the Air New Zealand nonstops between O'Hare and Auckland (if they are available again) and then connect immediately to a flight to Queenstown, and skipping visiting Christchurch entirely since you don't have time. Until recently there were many nonstops between Auckland and Queenstown, like hourly in the morning, to serve the overnight international arrivals.

Edit: You can visit Christchurch and the east coast of the South Island. Most people find the west coast more interesting. You likely will have to chose one or the other if visiting both islands, in your drive from Mt Cook to Picton to catch the ferry (or reverse direction if traveling north to south).

One of the prettiest of the South Island film locations is Deer Park Heights located about 10 minutes by car from central Queenstown: https://deerparkheights.co.nz/. It had been closed for a number of years but I see that it's now re-opened under new ownership. You'll recognize numerous scenes from the films, including the cliff that Aragorn goes over with the Warg during the battle on the way to Helm's deep (we're Ring Nerds too). The park itself makes for a pleasant day out. It sits about 500 ft. above Queenstown with sweeping views of both the city and the lake. There's a resident herd of nearly tame red deer that you can literaly feed by hand, and there used to be a variety of other creatures big and small to see from the confines of your rental car. Once at the top you're free to wander around and visit the various film locations at your leisure. It really is a neat place and well worth at least part of one day. Great place for a picnic too.
The town of Glenorchy just to the west of Queenstown is another significant film site, and not far from there are the trailheads for a couple of NZ's Great Walks if you feel like a bracing (mostly level) hike thru some of the prettiest back country on the South Island.

A few logistical suggestions: book a car with automatic transmission versus a manual, which like Europe tends to be the default choice. Even if you have experience driving on the left having an automatic will make it a much more pleasant driving experience for you.

You're wrong there @Robert . I finally bought an automatic car over 10 years ago - mainly because every cars I'd rented in NZ was an automatic and I decided I liked them. You would struggle to find a manual car for hire pre Covid - that will even more so now. Remember most of our cars are used imports from Japan and Koriea - they are all automatics