Hi Lisa!
I have not been to Prague and Budapest yet - hoping to visit Prague this Fall, however I can speak about Lucerne. I was in Switzerland last summer with my 18 yo nephew; it was at the end of our 2 week trip, already spending time in Rome, Florence, Milan, and Geneva - so we were looking forward to some down time (including cooling down, it was very hot in Italy last June!).
Lucerne was a nice place to relax for 3 days, although still managed to do quite a bit. We took the train to Bern one day on RS recommendation - we really liked it!
Another day we took the Gondola up and cog wheel train down Mt Pilatus - stunning views, I highly recommend it. Lucern - the city itself is nice, but overall was not that impressed; we spent evenings, and part of another day walking around town and seeing the sites RS has listed. It was very tough finding a room; it worked out but was also very expensive. We then went to Zurich for a few nights and flew home from there - RS doesn't give it much time but we went to the Zoo, and toured Lindt Chocolate Museum!
I have read many, many folks who said they were glad to have visited Berner Oberland/Lauterbrunnen Valley, that I think with your added interest of outdoor activities I think you would prefer it much more.
I have been to Vienna, really enjoyed it; don't miss the wine gardens! We took a short tram (15 min?) to outskirts of city and had a wonderful time; dinner and wine - shared table with others, a great memory.
I have also visited Salzburg, and Munich - loved them both as well.
I hope this helps! Good luck.
Maura
