What is the best way to travel between Munich and Vienna Austria?

The best service is Railjet, which is owned by OEBB, but you can buy it on the Deutsche Bahn website for the same price, and a bit more user friendly website. Buy nonrefundable, nonchangeable tickets 90 days in advance online for as low as 29 euro per person each way. 4 hour journey.
Edit-If you go by car, you need to get an Austrian vignette, which is like a toll sticker.
If you want to spend a day in Salzburg on the way, there are other rail options. Also, you can specify a stopover in Salzburg at no extra cost when you purchase, but you are locked in on your specific trains.
Stay very far away from Eurail.

"Best train to take? Germann Rail oe Eurrail?"

German Rail is the national rail company of Germany. Austrian Rail is the national rail company of Austria. Eurail is a cooperative effort of all the national railways in Europe (except British Rail) to package and market rail passes for Europe. Eurail does not run any trains.

The main train line toward Vienna is quite scenic, even without making any stops or detours, so you don't really need a car to enjoy some countryside. The link to German rail (above and on my earlier reply) provides the most complete train schedules for Europe and also happens to sell tickets for your route. Booking nonrefundable tickets up to three months in advance will give you the best prices. Four passengers can share the cost of a car rental, but booking ahead for a one- or two-day rental often costs the same as three days.

One other transport method you didn't mention, and it's definitely not the fastest and requires some physical exertion, is bicycle. Train stations in both Vienna and Salzburg rent great bikes that can be turned back in at any train station in Austria, if you cared to pedal along part of the Danube River, or to ride up to the salt mine at Hallein, outside of Salzburg. We did both and they were highlights of our trip! Happy travels.

I think the point that Eurail is a massive rip off travel agency has been covered.

Traveling from Munich to Salzburg and Salzburg to Vienna is best done by train. If you really want to explore around Salzburg, then maybe consider renting a car for a couple of days while there.

Traveling from Munich to Salzburg can be cheaply done by train on the Bayern Ticket. For 35 Euros all four of you (8.75 Euro each) can make this journey on regional trains. From Salzburg to Vienna, you have two options - OeBB or Westbahn. Personally, I recommend Westbahn as you can buy the tickets on the train and they have wifi. Tickets bought in advance on OeBB are 19 Euro each and Westbahn is 23.90 Euro each.

https://westbahn.at/en
http://www.oebb.at/en/index.jsp

I bet spending 55.40 euro per person for roundtrip travel Munich to Vienna is a better deal than Eurail!

Do you have 6 nights to tour Austria or 3 . . . I'm not clear whether the 3 in Munich at the end are included in the 7 or not. If you only have 3, you won't manage Vienna and Salzburg and countryside. If you have 6, then probably a car is best because it gives you the freedom to stop along the way for 1-3 nights between cities in Austria. Friends of mine spent a few days driving between them and enjoyed it very much.

I'd probably start with the Bayern ticket to get from Munich to Salzburg. Then, to see some countryside, rent a car and plan some brief stops on the way to Vienna. The biggest problem with a train is trying to make intermediate stops that aren't right on the line. If you aren't interested in getting too far off the tracks, there are plenty of trains to catch daily - just be sure the train you get on will stop where you want to get off and get back on the train (the faster trains may not stop there).

I agree with Brad about taking a train to Salzburg and picking up a rental car there to drive to Vienna. Then catch a train back to Munich.
Austria is a country that's best seen from behind the wheel of a car--especially with 4 travelers.

Too bad you don't have time to drive south of Salzburg down to Innsbruck and into the Austrian Alps. The scenery is beyond belief.

If you want to go slow and make stops, consider Austrian Rail's Einfach-Raus-Ticket It's similar to the Bayern-Ticket in Germany (unlimited regional trains in a day after 9 AM workdays) but for 2 to 5 travelers (not for single travelers) for 35€. Total travel time by regional trains on the main line between Salzburg and Vienna is about 6½ hours. At intermediate stations there are probably buses (fare not included in Einfach-Raus) to surrounding towns.

The second sentence in you post is: Car or train. Everyone seems to have gotten hung up on cheap trains. The answer to your complete question is a car, First of all. if you have read Rick's books, a car is generally cheaper for a party of more than 2. Beyond that you say you want to see the countryside and a car s the way to do that. Remember if you see something that rally looks interesting you can stop in a car and take a closer look, the train won't stop. Between Munich and Salzburg, Lidwig's island castle Herrenchiemsee is a very neat place to see. Lots of luck with a train, it can be done with inconvenience, but what to you do with your luggage???????? If you take a car it is easy to stay in the outskirts of a big city and take public transportation into the center to see it. Once you get to Vienna you want to get rid of the car a use the public transit.