I've lived close to the Austrian border for 49 years, and - of course - I've been there for many times.
First of all: Austria is VERY small. Apart from visiting the cities for cultural things, there's no need to change the places too often to reach the most interesting landscape sites.
A must-see in my opinion are the Krimml waterfalls (https://en.wikipedia.org/wiki/Krimml
). You can go up to the top, it's a nice and easy walk.
Another thing is walking alongside the river Lech in Voralberg, which is the most western part. You can hike for a few kilometers but also a few days. It's the last wildwater river up to the German border.
I wouldn't recommend to stay too close too Großglockner, it sounds like a tourist trap.
The northern and eastern part of Austria (close to Czech and Hungarian border) does not have that spectacular landscapes as the eastern and central part. Most spectacular mountains are Hohe Tauern. Also very beautiful: Ötztal (Tirol). But the most spectacular mountains (apart from Switzerland) are in South Tirol, which belongs to Italy...if you have time, have a look.
Survival tips for Austria:
DO NOT EXCEED THE SPEED LIMIT - no matter how fast the Austrians are driving. Otherwise you'll get some tickets, I promise.
Some foods and things do not have the same names as in German language (if you have a dictionary with you): Potato - Kartoffel (ger) - Erdapfel (aus). Prune - Zwetschge (ger) - Powidl (aus). Tomato - Tomate (ger) - Paradeiser (aus) and lots more.
Some recommendations to eat: Kaiserschmarr(e)n (kind of ripped sugared pancake), Marillenknödel (sweet apricot dumplings), Speckknödel (bacon dumplings - Tirol). And of course the most famous austrian sausage meal, maybe from a hut in Vienna: Eitrige mit Buckel und Schiß ("purulent one with hunch and poop"). It's a cheese-pieces-stuffed sausage with a bread roll (=hunch) and mustard. If the sausage is heated up, the cheese will melt, that's where the name comes from. Austria is very famous for black humor.
No, Stuttgart is not worth a visit - except if you're a fan of automobiles. If so, have a visit of the Daimler museum.