It looks like there are quite a few direct trains from VIE to Salzburg, more options changing in Vienna and Graz. That'll be by ÖBB (Austrian Federal Railway). From/back to Vienna, you also have the choice of Westbahn as well.

As for bags, you're likely going to have to haul them on and off the trains yourself. Book first (or even the more expensive business) class (cheaper - even quite affordable - if purchased in advance but inflexible - miss the specific train and you buy a new ticket) for more space. Not sure how Westbahn Plus compares.