Tips for traveling from Munich to Zermatt to Rome.

All drivers will need an International Driver's Permit, before you arrive in Europe. You will need a vignette to drive in Switzerland. And since you are dropping the car in a different country than you picked up, expect a very hefty drop off fee of up to €500 or more, in addition to the rental fee. Familiarize yourself with each country's driving laws and road signs. Respect the speed limits. Enforced by speed cameras, with a very small margin of grace. Recognize and avoid all ZTLs. Any infractions will result in a ticket that can arrive at your home months after your return.

Out of curiosity, is there a reason why you are choosing to drive rather than the often faster and cheaper option of trains?

Honestly I got overwhelmed trying to figure out how make this trip work with a train. When I tried to book a train from Munich to Zermatt, i was seeing multiple stops and connecting trains that seemed more difficult than just getting a car and just driving with a gps. I’ve read and been told it’s so much better and easier to just use trains but with all this being new to me, I honestly got overwhelmed. Any tips on making a train work for the three destinations we are trying to reach?

Trains are definitely your best bet for this itinerary and month of travel.
Seat 61 is a great place to get train info. Here's a start:
https://www.seat61.com/international-trains/trains-from-Munich.htm#Munich-Zurich

Completely understand why you got overwhelmed, but it is actually not that bad I think.
Munich-Zermatt will be some fast train from Munich to Zurich, and 1-2 slower ones after.
Zermatt to Rome is Zermatt-Visp-Milan-Rome, it seems.
Depending on how many people you are, trains can be much much or much less cheaper (A car in a quick check was about $ 2000, assuming two weeks, trains about $300 per person)

bear in mind that you are going to and through and around mountains in the middle of winter. The days are at their shortest, you will have your headlights on much of the time.

Maybe you'll get good views, maybe not.

The Matterhorn is moody at the best of times, often hiding behind clouds even on the best days - in mid winter there are no guarantees.

If the roads are clear when you arrive they may be snow and/or ice covered when you want to leave - you may wind up waiting for the snow plow.

Trains are designed for such weather. The high speed train from Milan to Rome goes at up to 325 km per hour. Your car is only allowed a max of 130 kph in some areas, slower in others, and tickets are handed out by speed averaging and instant cameras.

No way a car is faster from Zermatt to Rome.

Have you driven in Munich? It is no fun. As you drive towards Switzerland you will learn the meaning of Stau.

Don't be intimidated by trains. Watch a few youtubes, learn how they work. Way easier than flying, much more scenic, and even the driver (there is a paid person at the front, no work for any of you) gets the same views - no concentrating on the edges of the roads and traffic and corners and tunnels.

Your profile gives no hint where you live or how much winter driving experience you may have. So, how are you on winter roads and mountain driving?

Thank you for all the replies and help. I just created this account so definitely limited info. I am a truck driver living in Texas. I have a lot of experience driving in snow and ice while driving in winter Colorado and Utah. So I’m pretty comfortable driving in adverse conditions.

However I am taking all of your advice and sat down to really learn and figure out the train system and purchasing tickets, and I believe I have a better understanding of how it works. Another small thing I didn’t mention was luggage and initially not wanting to do trains because of having to carry around our belongings. My spouse LOVES to overpack. On top of that, we are going to be tourists and want to buy stuff to bring back home. Since I’m now leaning on train transport, I’ll have to talk to the group to make sure they don’t overpack and can get on a train comfortably with their belongings.

We also found a service called DayTrip where you can rent a driver to drive you to your desired locations. It’s more expensive but money isn’t an issue. Any feedback on renting such a service?

No experience with booking a driver, sorry. If you have a „luggage issue“ and like the flexibility, and money is not an issue - take the car. No reason to overthink it. If money is somewhat of an issue, do the math (and be aware that gas is much more expensive in Europe). Only you can decide how much the premium of unlimited flexibility and convenience is worth to you.