So where to stay? Anywhere near the old town. We were in the Rechte Altstadt (right old town) near Mirrabella Gardens and it was a lovely 10 minute walk down a cobblestone street to the river. We stayed in a semi budget hotel called NH Salzburg City - great location and I'd recommend it for the price.
Where to eat? Any wurst stand in town. At the old brewery Die Weisse for classic Austrian (one of the best meals I had on the trip). Wasserfall was a lovely change from all things germanic (Italian) and Die Alter Fuchs was great for atmosphere. The restaurant at the the Museum of Modern Art, M32, was outstanding. Get the lunch time special for a two course deal.
What do do? Wander around the old town and eat strudel, drink bad coffee, and take in the atmosphere and wonderful architecture. A great people watching spot is Carpe Diem - food looks good and the coffee was okay. There are some interesting galleries. The Rupertinum in the old town is worth a look in, and the walk up Monchsberg provides stunning views and the modern art gallery once you get up the top is amazing. Go in the morning and stay for lunch at M32. If you are around on a Thursday, try and get to the farmers market near Mirrabella car park. It was operating in deep snow and had great produce and amazing krapfen (jam donuts)
Whatever time of year you find yourself in Salzburg, you'll love it. A small town with enough to see - but not so much that you get to take your time and relax.