Answers of Kjel

You may be surprised how easy it is to travel around even with public transport in winter. Many good suggestions in the other comments.

I’d definitely travel onwards to Bern if that’s not included in your plans already. Liechtenstein is Nice if you want to tick a country, but not that scenic. St Gallen is beautiful and walkable as well.

Actually traveling straight to Lucerne would be smart and take day trips from there is doable (to Bern for example).