Yes, Bern is very realistic - only 1 hour on the train. Same for Basel, although I would prioritize Bern.
For day trips, you will have plenty of options (and hopefully you have a train pass!). You can easily get to a variety of the nearby mountain areas (Engelberg/Titlis, Pilatus, Rigi, etc). With good weather, also try to incorporate a lake cruise from Lucerne - you can do this as part of a visit to some of the mountains or as a stand-alone activity and the "Half Tax" or Swiss Travel Pass train cards are valid.
Something like Wengen or Zermatt is a bit far (+- 3 hrs each way) for a day trip, but if you just want a nice train ride with scenery you could make it to Interlaken and back easily (2hrs each way, no changes) or get off along the route (Meiringen for example, visit Reichenbach falls, the Aareschlucht).
Other possible day trip options if you want to see the Italian-speaking region of Switzerland would be Bellinzona (about 1 1/2 hours each way, UNESCO world heritage site with castles) or even Lugano or Locarno (about 2 hrs).