Travel to Germany and Austria in June 2017

We wish to see a little of each country so I am thinking somewhere between Munich and Salzburg, changing hotels once or not at all if that is doable.

Certainly, putting yourself in SE (Upper) Bavaria will allow you to see "a little of each country". But just be aware, if your goal was to compare and contrast the two, Upper Bavaria and the adjacent Austrian regions of Upper Austria, Salzburgerland and Tirol are the areas in which the two countries most closely resemble each other. There's plenty to see and do in the area, you just won't get a sense of the vast diversity of either country (moreso Germany). If this was my trip, and I was only using public transportation, I would probably stay in either Salzburg, Munich or Rosenheim. If by car, perhaps Berchtesgaden, or Prien am Chiemsee.

Whether or not to rent a car or travel by train depends on your travel goals. Trains make the most sense if traveling between cities, particularly large ones. Although you certainly can explore the countryside via public transportation, it isn't the most convenient method.

Plenty of guidebooks will provide good recommendations for southern Bavaria/western Austria, so rather than repeat the obvious ones, allow me to suggest a few less well-known locations.

Burghausen, on the Bavarian/Austrian border. It has a nice colorful Altstadt (old town), but the main reason to visit is the castle. Depending on one's definition of the word "castle", you can consider it one of the biggest in Europe.

Landshut also has a castle, but a very large and extremely colorful Altstadt. I visited Salzburg soon after seeing Landshut, and I can say that the rather small, monochromatic historic core of the former did not compare well to the latter, despite attracting orders of magnitude more tourists.

Passau, because... well, do a google image search to see why.

"I am thinking somewhere between Munich and Salzburg..."

Prien is right in between, an hour from each by direct train. Prien is home to Herrenchiemsee Palace (on a nearby island in Lake Chiemsee) and 15 minutes by train from Aschau, where you can catch a lift to the Kampenwand for a walk/hike and some scenery.

https://c1.staticflickr.com/7/6177/6139253776_7b61f5d359_z.jpg
http://0.www.alpen-guide.de/m/image/db/159193_luftbild-herrenchiemsee.jpeg
http://www.schloesser.bayern.de/englisch/palace/objects/hch_ns.htm

Lift from Aschau: http://www.chiemgauresort.de/medien/kampenwandbahn.jpg
Mountain map: http://kampenwand.de/index.php?rex_resize=835h__panoramakarte_sommer.jpg
Sonnenalm terrace, roughly 100 yards from the lift: https://media-cdn.tripadvisor.com/media/photo-s/0b/77/9b/a2/sonnenalm-kampenwand.jpg

"Is there a better location/city to travel between the countries?... I love the food, the small towns and the mountains."

I think Berchtesgaden is very nice as well. Easy bus trip into Salzburg. Not so close to Munich however.

Outside Salzburg: You may wish to visit Hohenwerfen Fortress and Falconry Center in Werfen:
http://www.markmarissink.com/wp-content/gallery/hohenwerfen/DSCF0697.jpg
http://www.salzburg-burgen.at/en/werfen/erlebnisburg-hohenwerfen/index.htm

With 10 days and your interests, I would suggest 6-7 days or so in the above area and 3-4 days in the Rhine/Mosel region as well, where you'll find dozens of attractive old-world villages and a crop of castles from medieval times. It's about 5 hours by train from Munich to this very scenic area:

Rheinstein Castle: https://www.uli-franke.de/wp-content/uploads/2014/07/20140622_144833-Burg-Rheinstein.jpg
Rheinfels Castle: https://www.youtube.com/watch?v=qxFF80wORNQ
Day cruise near Bingen: http://www.bingen-ruedesheimer.de/wp-content/uploads/2012/10/LZ1B0487-als-Smart-Objekt-1.jpg
St. Goarshausen: http://www.staedte-fotos.de/1024/sankt-goarshausen-altstadt-burg-katz-23553.jpg
Bacharach: https://upload.wikimedia.org/wikipedia/commons/4/4c/Bacharach_BW_9.JPG
Cochem: http://www.der-takt.de/typo3temp/_processed_/csm_NEU_T_Reichsburg_Cochem_3_087e83a8d1.jpg

Cochem's Mosel Wine Week takes place June 14-18... wine stands, wine queens, street music, fireworks...
http://www.cochem.de/tourismus/bilder/seiten/bilder_Bild_177.jpg

It would make sense to fly into Munich and out of Frankfurt (or vice-versa) for an itinerary like this one. Trains serve both areas well, and of course you can drive too.

I'd suggest flying into Munich and stay in a hotel in the center city. Munich is a great city, and there's enough tourist sites and beer to last at least 4 days. And don't miss Dachau while you're there.
Then take a train down to Innsbruck, and pick up a rental car for the remainder of your trip. Day trips out of there are glorious in mountain scenery: The Austrian Alps are about the most beautiful place I've ever experienced.
Salzburg is a good day trip from Innsbruck.
You could go east to Zell-am-Zee and get on the Grossglockner High Alpine Highway that takes you over the second highest peak in Europe. You end up just over the Italian border where you can go west to the motorway that brings you back into Innsbruck through the Brenner Pass.
Fuessen is close by, as is Crazy King Ludwig's Neuschwanstein.
We love getting up on the mountain in Western Tirol and just rambling from village to village by car.

If you had another week, I'd tell you to head for Hallstadt and Vienna.

Wonderful ideas! I cannot wait to get home and look at the places you have suggested. Looking on a phone leaves a lot to be desired. We are excited to begin planning.

We will do air, then hotels/houses and then the fun begins, picking things to do!