A couple more thoughts.
Weather is quite dependent on altitude.
I was in Switzerland a week or so ago, and everywhere below 800 meters , with an exception in part of the Emmental, and a day or so in Zürich, there was fog all day long most days. That pattern is common in Winter, but the fog usually burns off by midday.
However, if I got above 600 meters in the Emmental, or at high altitudes near Wengen, there was clear, sunny weather.
Since you will be in Bern, you might want to visit the heart of rural Switzerland, the Emmental. Nice hiking. Your cousin will likely know more about it.
If not, here a thread with some information:
This map is useful. If you work the menus, you can get displays of all the transport stops, etc. Zoom in far enough and the rail lines show up:
For a three or four week stay, you need not limit yourself to day trips from Bern. Use the trains; you'll be impressed.
There are inexpensive accommodations all around Switzerland. Many are oriented toward backpackers and may have dormitory accommodations ( "Matratzanlager"). for the lowest cost options.
Youth hostels are very high quality.
Some good links there.
Under "AirBNB," read about alternatives at the bottom of the page.
Do you speak any non-English language?