Stuttgart, Germany

I would like to find some beautiful places to visit during the week
that are not too far from Stuttgart and safe for me to venture out on
my own,

There are many, some on the S-Bahn and some that require a longer trip. Ludwigsburg a bit north of Stuttgart e.g. or Esslingen. A bit further away cities like Ulm, Mannheim and Strasbourg are all on day trip distance.

Best way to get to Sardinia, Amalfi Coast?

Define best? Cheapest? Fastest? Most scenic?

I think fast, scenic but we don't want to spend a fortune. We are good at doing things cheap. I don't mind trains but they do take longer. I think it might be quicker to fly. My husband has some family ties to Sardinia so he wants to go see it. We have so many places on our bucket list, Prague, Pompeii, Crete, Santorini, Sweden, I have a friend there. Norway too! But we have to do one at a time. He will have Friday afternoon of for sure and would go back to work on Monday.

Eurowings has flights from Stuttgart to Naples and Olbia. But a train trips across the Alps will give you some great scenery.

Don’t know when you’ll be in Stuttgart, but their Christmas market this year begins Nov 24 and runs till Dec 23. As far as day trips, your best option by direct train are Heidelberg and Mannheim (45-minutes).
You can fly non-stop from Stuttgart to Olbia as Badger said but it looks like twice a week. The same with Olbia to Naples, but I see four days a week for non-stop flights. You can then work your way to the Amalfi Coast from Naples by either ferry or train.

If these are all weekend trips from Stuttgart to Italy, yes, you’re going to need to fly, and you won’t get to see much at any of your destinations.

I don’t know what Italian rules currently are for Americans visiting from Germany; also, your vaccine status will make a difference, almost surely.

Maybe you explore and try finding the pros and cons of different areas during the week and hubby comes to the cream of the crop on the weekends?

Cross border covid restrictions change frequently, both for the better and for the worse... keep abreast of them and adjust according. Prepare to be flexible.

The storms were no issue for hubby it sounds.

If you hope to move this winter I guess that your visa is well on the way? Will he need to be near Stuttgart?

If you branch out to Switzerland and Germany after Italy you will find that Stuttgart is right on the doorstep of absolutely tons of fabulous places, but again with the severe damage from the storms be sure to check before journeying to the Alps or down the Rhine.

Some of the places that you listed seem to be incompatible with your diet. Sardinia, for example, will have a lot of fish and crustaceans and octopus, and garlic is is in many if not most of the traditional dishes. Goat cheese will be popular.

Onion is in many German dishes. Onion is Zwiebel in German.

If you are in Stuttgart you are in automobile mecca..... Suggest visiting the Porsche Museum in Zuffenhausen home of the 911.... Even if you aren't a Porsche fanatic, it is a cool museum.