The Sound of Music was filmed on Soundstage 15 at 20th Century Fox and on location in Austria and Germany. The principal cast and crew were in Austria for a total of 11 weeks - it was supposed to be 6 but the rainy weather delayed the film. Luckily it more than made up for the extra costs.
Here's the best tour site I can find with additional information for those who are actually traveling to Salzburg, Austria.
Here's a map of all the locations.
This site has a step-by-step walk-through if you want to get really technical. Also, on the Blu-ray special features disc of the 45th and 50th anniversary sets is a fantastic interactive map of all of the locations used with film clips, additional facts, and photos. I highly recommend looking at it.
Anif Castle, Salzburg - seen in the opening aerial shots
Schloss Fuschl, seen in opening aerial shots. Located about 40 minutes outside of Salzburg.
The field, which is private property, is located in Mehlweg, Germany, about 6 miles from Salzburg.
The stream and birch trees were brought in for the scene. An angry farmer, on whose property it was built on, punctured holes in the stream with his pitchfork, halting production while it was fixed.
The abbey gate seen in the film is from the Nonnberg Abbey where the real Maria Von Trap was a postulant. The nuns there allowed it to be used for the film.
Another exterior shot of Nonnberg Abbey (you can see the changing heights of the children here as they grew).
Schloss Frohnburg Palace, used for the front and back of the Von Trapp Home
It is located just off of Hellbrunner Allee, an alleyway that connects the Hellbrun Palace with Salzburg.
Hellbrun Palace is also where the original gazebo from "16 Going on 17" is located. It was built at Leopoldskron specifically for the film. It was moved to Hellbrunn so that tourists could visit it. The interior shots of the gazebo were filmed on a sound stage, as the real gazebo was too small for the dance scene. No one is now allowed to go inside because some old lady tried to do the dance and hurt herself... yeah.
The front of the movie house, Schloss Frohnburg Palace (belonged to actress Hedy Lamarr at the time).
The real Von Trapp home. It was taken over by the Nazi's and used as offices during WWII. It is now a hotel.
The Mirror Room inside Schloss Leopoldskron in Salzburg, which was copied and used as the ballroom.
Schloss Leopoldskron, Salzburg, Austria. Used as the back yard of the von Trapp house (actually they built a replica of this on the lot next door for filming).
The terrace at Scloss Leopoldskron where Uncle Max has pink lemonade.
The "Do-Re-Mi" sequence was shot in nine different locations:
The Mozart Footbridge (Mozartsteg crossing the Salzach River)
The hills of Werfen (Untersberg Peak can be seen from their terrace). Also the meadow beneath the Untersberg.
Winkler Terrece
Mirabell Gardens
The Pegasus Fountain in Mirabell Gardens
Mirabell Gardens Tunnel
Mirabell Garden steps, from a different angle.
Statues in Mirabell Gardens
St. Wolfgang Schafbergbahn Mountain Railway (the final scene filmed with the children on location).
Pferdschwemme (Horse Pond), Karajan Square - a scene that was cut from the movie was shot here, more about that in the post on Charmian Carr's book, Forever Liesl.
I think this is where the children climbed the trees but I can't find where I found the photo.
Bertelsmann on Leopoldskron Lake, where the children were filmed boating.
Bertelsmann from the air - I think it's also in the aerial shots at the beginning.
Mondsee Church (mond-say), where the movie couple were married. It is located in the village of Mondsee, which is situated in the middle of a lake at the base of sheer cliffs. It is about 20 minutes from Salzburg. The real Maria was married at Nonnberg Abbey.
The exterior of the church.
The Residenz Platz (Residence Square), where the Nazi soldiers are seen marching. When that scene was filmed, the town did not allow the public to watch. Also where Maria sings "I Have Confidence" near the beginning of the film.
Felsenreitschule (Rock Riding School), Salzburg - film location for the final performance of the von Trapps. This is the stage that the real Salzburg Festival takes place on every year.
St. Peter's Cemetery
Though not actually filmed here, the St. Peter's Cemetery was recreated on the sound stage at Fox so it's still a cool place to visit. It's located in the St. Peter's District of Salzburg.
The final scene of the movie shows the von Trapps hiking over the mountains into Switzerland (which is geographically and historically inaccurate - they really just drove a few minutes to the border). This scene was filmed at Mount Obersalzburg in Rossfeld, just a little ways from where Hitler's "Eagle's Nest" at Berchteschgaden, was located, giving the cast and crew a chilling reminder of how near the danger really was. It is a mere 12 miles from Salzburg.
Sources:
The Sound of Music Companion by Lauence Maslon, 2006.
Forever Liesl: A Memoir of 'The Sound of Music' by Charmian Carr, 2000.
Sound-of-Music.com
Images found via Pinterest
More Travel Links:
Panorama Tours
BigBoyTravel - SoM Movie Locations -
Map
Salzburg Official Site
Map of Locations
I will also be adding a new feature: Netflix Monthly Movies, where I will review a classic movie currently on Netflix Instant. Most of the movies on Netflix seem to be one's that aren't well known and have actors with names no one really remembers anymore. This doesn't always mean the movie is bad however. And a lot of times, there is no trailer on YouTube either, so the only way to know if the movie is good or not is to watch it. My first review will be this coming Monday: The Flight That Disappeared (1961). The Blonde at the Film also posts about the more famous classic movies currently on Netflix Instant. 
