Confused About Public Transportation

An important information is missing: Does your planned stay include a change of calendar month? Likely not based on your input.

The Deutschland-Ticket is a calendar monthly subscription tariff to use all inner-German regional trains and local public transports incl. Salzburg station. Therefore a stay including two different calendar months would mean "buying" two months per person. Never forget to cancel the subscription early enough to avoid automatically booking the following month.

An alternative - not for Stuttgart journey - is the Bavaria Ticket.

Since "smartest choice" is no objective criteria I like to return questions to you which might help you to find your answer:

1. Question 1 is standing in first line of this post
2. How much / often are you willing to book tickets?
3. Are you willing to use mostly regional trains? Tip: compare train travel times to destinations (https://int.bahn.de/en).
4. How relevant is budget for you?

btw: you could do a car round trip of all this with overnights not in Munich. German Alpine Road is a scenic alternative. You might need an IDP for this (see regulation). A German registered car needs a vignette for Austrian motorways.

"Question 1 is standing in first line of this post
How much / often are you willing to book tickets?
Are you willing to use mostly regional trains? Tip: compare train travel times to destinations (https://int.bahn.de/en).
How relevant is budget for you?
btw: you could do a car round trip of all this with overnights not in Munich. German Alpine Road is a scenic alternative. You might need an IDP for this (see regulation). A German registered car needs a vignette for Austrian motorways."

We are going from August 16-25. I feel like the Deutschland Ticket would be so much easier because how much time do I really want to spend booking a lot of tickets. I think I would be fine with using mostly regional trains as I do want to keep an eye on a budget.

I realize now it probably would have made more sense to not stay in Munich the whole time and did a round trip with overnights. It's too late to decide to make that change now.

Regional will work well enough for Neuschwanstein, Nuremberg and Salzburg. Innsbruck and Stuttgart are 3-4 hours each way with regional, so rather prohibitive. High speed is about 2 hours, and depending on the train you choose would be from 33€ per person (if booked for a specific train). A cheap and clean hotel in Stuttgart like B&B Stuttgart City would be 100€ for all, so you might be cheaper off paying the additional night and going regional. Innsbruck is not covered by Deutschland Ticket anyway - but there are plenty of good alternatives within reach (though which is best depends on what you like). One I directly thought about was the medieval island town of Lindau on lake Constance, where you might even be able to take a swim, or make a cruise on the lake, including a bit of Austria and Switzerland (no problem passport wise - all are Schengen)

The Deutschlandticket would make sense, as it allows you to use all public transport throughout Munich. For navigating Munich, I would recommend the MVgo App.

You could also use the Deutschlandticket to take trips to Augsburg or Nuremberg for example. I really like the website Chronotrains because you enter where you're departing from by train and how long you would want to travel (I chose Munich as my base, with a travel time of 2 hours): https://www.chronotrains.com/en/station/2867714-Munich?maxTime=2
And to find out whether you can still use the Deutschlandticket when you leave Germany, enter your desired route on the DB website and check the Deutschlandticket box beforehand - https://int.bahn.de/en

Otherwise, there's plenty to do in Munich in the summer. I'll link you to a few pages, but one thing's for sure... don't forget your swimsuit (the Isar River is so clean even in the city center that you can even swim there, but of course there are countless other possibilities)
https://www.munich.travel/en/categories/discover/urban-districts/summer-in-munich
https://www.munich.travel/en/topics/things-to-do/the-best-tips-for-this-summer

Day trips from Munich: https://www.munich.travel/en/topics/lakes-mountains/summer-excursions-munich

And I would definitely recommend that your teenagers stop by 'Bahnwärter Thiel, the 'Alte Utting,' and the 'Werksviertel' behind the Ostbahnhof. Munich isn't just beer and lederhosen, it's also street art and young. I probably don't need to mention the surfers in the English Garden anyway :-)
https://www.munich.travel/en/topics/urban-districts/local-love-munich/local-love-schlachthofviertel-the-charm-of-the-casual
https://www.munich.travel/en/topics/convention-bureau/news-highlights-2024/the-werksviertel-in-seven-tips

Assuming that your travel takes place during one month, IMO the D-Ticket is going to be the most economical and flexible. It covers all of your local/regional trains and all of your public metros, trams, and busses. Only exception is the "maybe" to Innsbruck, though it does cover the cost to and from the German border. It is also flexible in that you don't need to book ahead or be tied to a specific train schedule in order to get the best priced ticket. At 58 divided by 8 you only need to spend 7.25 per day to break-even.

I realize now it probably would have made more sense to not stay in Munich the whole time and did a round trip with overnights. It's too late to decide to make that change now.

About your day trip plans... if you keep them all...

Salzburg: 3.5 hours getting there and back by train, so you should have a firm schedule for seeing whatever it is you wish to see there. With the D-Ticket (a solid idea as the Bayern ticket has an after-9-am rule on weekdays!) you can leave early and come back late for this day trip. Replacement ideas:

Nuremberg: Also 3.5+ hours round trip. Same strategy as for Salzburg - go on Sat or Sun.

Stuttgart: Forget Mr. D-Ticket unless you want to waste 7-8 hours on the train. Buy normal tickets for the high-speed trains. This will still require 4-5 hours of train travel, however.

Innsbruck: 5+ hours on this day with the D-Ticket. Get normal tickets for the high-speed trains. Maybe 3.5 - 4 train hours now.

Neuschwanstein: Around 5 hours round trip on trains/buses w/ the D-Ticket for this 25-minute tour.

They say there's no such thing as bad weather if you change into the right clothing... There's also no such thing as a bad day trip if you are willing to change to more-doable outings... I suggest dumping the Innsbruck and Neuschwanstein outings and replacing them...

• Day trip to Kloster Andechs: (see Rick Steves' guidebook.) Less than 1 hour to Herrsching on the S-Bahn + a nice hike. Or take the train + bus connection directly from Munich. The hike is well marked and a popular way to get there as the scenery overlooking Ammersee Lake is super. Interesting monastery, fun food, good beer, family atmosphere. https://www.andechs.de/en/visitor-information.html

• Schloss Nymphenburg (Nymphenburg Palace): Much more interesting than Schloss Neuschwanstein, and right in Munich. King Ludwig II (the Bavarian King who built N'stein) lived in this place as a kid. You'll need 2+ hours to see/do all there is to see/do.

https://www.theworldisabook.com/16991/munich-with-kids-nymphenburg-palace/

https://schloss-nymphenburg.de/ (choose english)

The DB ticket will work, but it's not ideal for a day trip to Stuttgart. You want the ICE for that or you'll spend the major part of the day just riding the train there and back. And if a train gets delayed your trip is going to get much longer. I'm a Deutschland ticket holder, but when I go to Munich, about every six months or so, I buy a seat on a fast train. The ICE takes less than 2 hours. And, if you use the DB app, it doesn't show you a direct route, you have to know how to use the regional lines to get between those two cities. I have done that, but it's only worth doing if you plan to stop along the way.

A lot of the other stuff can be done on the Bavaria ticket; Salzburg, Nuremberg, public in town transit. Would that be better? It's going to depend on what your day to day plans are.

I only spent 5 night in Munich with no day trips. If you definitely pick where you are going and then just buy the few round trip tickets you need, you have no randomness or uncertainty and it takes less effort than trying to figure out whether it is worth buying the ticket that lets you get on and off trains randomly. Check google maps and the Deutsche ban website. Your 5 supposed day trips are too far from Munich to see properly as day trips. Are you sure it wouldn't be wiser to delete 3 or more nights from Munich and add one or more nights in Nuremberg or one of your other 4 locations? How many night total is your trip?