You don't mention too many of your interests except that they don't include skiing on this trip. If you tell us more about yourself, it will help people give you suggestions.
Six days is hardly enough to scratch the surface in Switzerland so I would not try to do Austria unless there is a "must see sight" there that you don't think you can live without seeing.
In general, people fly into Zurich and then move onto their first location that same day. Luzern is a popular first stop as it is not far by train (you get the train right at the airport) and it has a variety of activities involving both lakes and mountains. You could spend two nights there. I would move on to the Berner Oberland area which is a favorite location for many of us on this forum. I would spend 3 nights there. I'm not sure how much snow is left at the end of May, but I usually stay in Murren. Lauterbrunnen is lower and might be a better choice. Hopefully, someone who has visited in May will be able to tell you.
That leaves you one night on the way back to Zurich. You might enjoy visiting Bern.
The more information you can post here about your interests, the better answers you will get.