AUSTRIA/Switzerland

I'd keep it simple ...
Vienna (2)
Innsbruck or Salzburg (2)
Luzern or Lauterbrunnen (2)
Geneva (1)

If traveling by car, be aware that dropping the car in Switzerland would come with large drop off fees. You could drop it in Western Tirol and take the train on to Switzerland.

Thanks for your suggestion.
What would be the best means of transportation between these points and do you have any suggestions for accessing the slopes at these locations.
Thanks again

Train would be my choice.

Are you expecting skiing in September?

If you want to spend time in the mountains, skip Innsbruck as it is a medium size city in a deep valley surrounded by mountains, but not in them. I'd do St Anton as the lifts will still be operating for Summer hiking. It is right on the main line railway between Vienna and Zurich with direct trains every two hours.
Leaving there, the train continues into Switzerland. I'd do either Lauterberunnen or Zermatt.
But if you want to see Vienna, Salzburg, and spend some quality time in the mountains, in ONE week, I think you might have to just see Switzerland from the train window.

Thanks guys for the replies.
I hope you were being ironic, Nigel.
I did find this lovely site with fares and schedules and other info for the Hungerburg funicular in Innsbruck. (http://nordkette.com/en/)
and I was hoping for something comparable for St Anton because that seemed like a very attractive possibility in Innsbruck.
What do you think?

no, I wasn't being ironic.

We get all sorts of questions from all sorts of people and some people are more informed than others.

The question just up-thread was

any suggestions for accessing the slopes at these locations.

Which many people would interpret as looking for snow, skiing or snowboarding.

I haven't heard of looking for access to slopes in regards to hiking.

Sorry, Nigel.
Just didn't know how to express my interest which is in not-too-rigorous-hiking because I'm not sure about my endurance. So assistance in getting up the mountain or "slope" in the form of a funicular, for example, is appreciated.
Thanks for your engaging.

The places in David in Brisbane's answer at the top - I'd take the train.

He suggested

Vienna (2)

Innsbruck or Salzburg (2)

Luzern or Lauterbrunnen (2)

Geneva (1)

In all of those places there is plenty of easy local transportation and easy connections between them by train.

At Salzburg there is a funicular which goes up to the old Fortress and you can walk all around and into the woods, and all the way across take an elevator, and electric buses link you into the rest of the town.

Innsbruck is in the valley (named for the bridge over the river Inn) but the train goes right up the cliff and there are plenty of walks along and at the top, or you can stay on the train to Garmisch Partenkirchen or Mittenwald where you can walk in the woods or ride a cable car or cog railway up a mountain. Or walk along the Inn river down in the valley.

Luzern train station is right adjacent to the Altstadt (old medieval city) and the docks for the lake steamers. Board one of them and they will take you to all sorts of places where you can go up in a chair lift or cable car or gondola and hike up in the woods, or you can ride a train a short distance to a cog railway or a bus to a cable car, and often be up in the snow most of the year.

Lauterbrunnen is in a high valley of the Aare river, the same one which flows through Bern (which is why the river float in Bern is so cold - it is all glacier water just melted) and up from Lauterbrunnen is everything you could imagine, and much that you wouldn't imagine, about being at the top of the Alps. And you can hike along the river in the Lauterbrunnen Valley.

Geneva is big and pretty flat. There are places to hike and walk not too far away, both along the lake in Switzerland and across it in France. Evian water comes from the French side.

What great information from Nigel.
I was also unsure of the reference to "slopes" when travelling in September, but assumed you meant for hiking. An excursion to Pilatus from Luzern may be of appeal.
http://www.pilatus.ch/en/

For St Anton: http://www.stantonamarlberg.com/en/summer

So thanks so much. You all are great.
I think we've decided to do Vienna, St. Anton, Lauterbrunnen, & Geneva, 2,2,2,1, as suggested.
We're now looking at train schedules, but the St Anton--Lauterbrunnen segment is a killer.
Thanks for the St Anton site, Sam. It's exactly what I was looking for.
I'll keep you posted. Oh and if you all have any hotel recommendations,don't hesitate to pass them on.
Also we're not a 100% on Lauterbrunnen versus Wegen?

"Also we're not a 100% on Lauterbrunnen versus Wegen?"

You will find fans of all four places to stay in the Berner Oberland - Wengen (note spelling), Lauterbrunnen, Mürren, and Gimmelwald. I stayed in Mürren and was very happy with this. I would definitely choose Mürren over Lauterbrunnen or Gimmelwald; I didn't see Wengen so I can't compare it to the others.

But you really can't go wrong - any of the four where you find suitable accommodation for you will work well. The key thing is not to stay in Interlaken, which is near but not in the mountains.

Wengen is a car-free village and very convenient if the Jungfraujoch excursion is something you are thinking about. But when it comes to Swiss charm, it doesn't hold a candle to Muerren.
PM me if you are looking for a nice inexpensive B&B in the center of St Anton.

Edit - The train from St Anton to Lauterbrunnen is not quite as daunting as it seems at first look.
Its a direct train to Zurich, then you need to get to Interlaken Ost, and there are no direct trains, but require at least one connection. The fastest way is via Bern, but the way via Luzern is more scenic over the Brunig Pass on meter gauge railway. At Interlaken Ost, you change to the Berner Oberland Bahn to Lauterbrunnen. Now you need to be sure to board a car going to Lauterbrunnen as the train splits half way, and the other half goes to Grindelwald. Platform staff will direct you to the right part of the train, they've been doing this for a while.

At Lauterbrunnen, your options are to stay there, or go to Wengen or Muerren. Wengen is a simple train change. There are two ways to get to Muerren. One is to take a short gondola ride from the station up to Gruetschalp where you get on another train for a short ride to Muerren. The other way is to take the postbus to Stechelberg and then ride the cable car up to Muerren, passing through Gimmelwald.

Best train connections for you from St Anton to Lauterbrunnen.

RJ 362
Dep 11:02 ST. ANTON AM ARLBERG (Austria)
Arr 13:20 ZUERICH HB (Switzerland)
IC 720
Dep 13:32 ZUERICH HB (Switzerland)
Arr 14:28 BERN (Switzerland)
IC 1073
Dep 14:34 BERN (Switzerland)
Arr 15:28 INTERLAKEN OST (Switzerland)
RE 173
Dep 15:35 INTERLAKEN OST (Switzerland)
Arr 15:55 LAUTERBRUNNEN (Switzerland)

I appreciate your efforts

Thanks yoy all for your generous suggestions. Something came up and the trip was made impossible. Perhaps in the future....