Home base for our trip

Is Zürich your only option ? Munich is much closer to what you want to see, especially without a car.
Also, are you really set on Zugspitze and that area? It is beautiful, but there are plenty of more dramatic Alpine settings nearer Zürich if that's where you have to land.

Going to your question: Without a car, it seems tricky to just have one base. Garmisch-Partenkirchen would work for everything (EDIT: I initially thought it would be difficult for Neuschwanstein, but apparently there is a good train+bus connection).
Munich is definitely worth a day as well.
Lastly, March is ski / snow season still. Zugspitze will probably be COLD, Partnach gorge could be snowy (although with climate change, there is less and less snow at Garmisch's elevation).

I agree with balso on a lot of points. Have you checked the status of the castle at that time of year? Check their website - only guided tours at this time, but March could be tricky with snow/ice. I stayed in Mittenwald for several days, but it was summer (at least 10 degrees cooler than anywhere in Europe). There’s a small museum for the art of violin making. With restrictions - who knows. I had a great visit to Innsbruck. There’s a nice cultural museum with well preserved implements. If your a cook, it can’t be beat for old cooking utensils.
Lots to do in Munich.
I’m a fan of Zurich with its lakeside and historical center and ease of getting to other places by train. Have you been to Lucerne? Always a great choice but beware of high prices.
Good luck!

Balso has made some very good points. Munich makes much more sense as an arrival point. If cost is a factor, weigh on the added cost of getting to Garmisch from Zurich, and all the time you lose in doing so.

Garmisch-Partenkirchen is the logical choice as a base for the places you want to visit. Read up on the current restrictions in place for hiking the gorge, and be aware that it can be closed at short notice.

Innsbruck can be reached by train or bus. Neuschwanstein is more difficult without a car. You'd have to take a train or bus to Fussen, then a bus to the castles.

Early spring has unpredictable weather, and both roads and train lines can be closed due to avalanches or the threat of them, necessitating sometimes lengthy detours.

OP
Check out a recent post on Switzerland and Zurich.

Garmisch, for sure

We are open to other places as well.

"Other" places, meaning "in addition to" your stated destinations, or "in lieu of" those?? I will only look at the area you have targeted here, but there are other options in Germany that are closer to Zurich.

Garmisch-Partenkirchen: a good base for the Partnachklamm and the Zugspitze, but it's at least 5.75 hrs. from ZRH airport by train.

INNSBRUCK: The train trip from ZRH airport to G-P is nearly 6 hours, but Innsbruck can be reached in less than 4 hours with one change of train (in Zurich.) You should probably go to Innsbruck first.

After a stay there, you can move on by train to Garmisch-Partenkirchen (1.3 hrs. from Innsbruck by direct train) for a few nights. G-P is a good base town for ALSO for MITTENWALD (20 min. by train,) which you should not miss if you like mountain lifts, river gorges, Alpine atmosphere.

It is incorrect to say that a day trip from G-P to Füssen (for N'stein) and back is not feasible. G-P > Füssen by train + bus takes just 1.3 hours - then you catch the shuttle to the ticket center. Many people make the day trip from Munich to Füssen and back by train - which takes a full 2.0 hours in each direction.

(The train ride between G-P and Reutte is extra-scenic, btw.) The tour of N'stein takes only 30 minutes. You'll have plenty of time left in your day.

I was unaware of that bus connection from Garmisch to Füssen, so I stand corrected and will edit my post accordingly! Thank you

Two questions:

• "7 days", does that mean seven full days in the country, or does that include arrival and departure days? It's usually easier to count the nights you have: 7 nights will give you 6 full days.

• Like some of the other contributors, I would like to ask the obvious question "why Zürich"? I'm not saying that Zürich isn't a worthwhile destination, but all your other destinations are somewhat close together, with the exception of Zürich. Given your limited time frame, I would try to avoid spending too much time on the road resp. on the train.