Apr 27, 2017 - 1:49 PM
Luzern and Interlaken both have good train connections, although there are more trains departing from Luzern. You can check the timetable for the destinations you wish to travel to.
The Bernina Express runs in another part of the country. A very long day trip from Luzern is possible as explained here. I'm not sure if you'd like to spend that much time on trains in one day. There are other scenic train rides much closer to Luzern and Interlaken too, like the Golden Pass line, and mountain trains to Mount Rigi and Kleine Scheidegg.
Note that there is plenty to see for weeks just around Luzern or Interlaken. So you won't run out of options with 3.5 days to spend there. Yes, Zermatt is wonderful too, but it takes 2 - 2.5 hours to get there from Interlaken (one way) and even more from Luzern. The various links in my and Lucas's post provide more details so you can decide if the traveling time is worth it to you. Personally I would explore one region, considering the limited amount of time and the fact you'll be back for more later.
Dogs are allowed on cable cars. On Sundays most stores will be closed but in larger towns like Interlaken and Luzern you'll find some open ones too. Next Monday is Tag der Arbeit but grocery shops will mostly be open as usual.
Once you are sure about your plans you can pick a train pass. The Swiss Travel Pass will be worth it if you expect to do quite some longer trips. Otherwise the Swiss Half Fare Card, Tell-Pass or Regional-Pass Bernese Oberland will be better.