Thun is very beautiful in May, if you choose it as a destination then I recommend this spring promotion. I purchase it every year - you can do three of the 5 experiences and I always choose the three mountain excursions. These mountains are around 60.00 each to travel up and down again, so 59.00 for three is a steal! And there are really no crowds yet at this time of year.
Stockhorn mountain has a wonderful one hour walk around the lake at the mid-station, quite easy and often snow free, or mostly so, at this time of year. The views from all three mountains are pretty incredible.
You can also choose to ride on the lake, see the St.Beatus Caves, eat lunch there, head up the Niederhorn - all in the same area, minimal additional expense for the bus, can also choose to return to Thun by boat. To reach the Stockhorn and Niesen you need to travel by train, but for around 40 minutes or so.
If you are in Thun on a Wednesday or Saturday you can walk through the market. Sometimes you find authentic crafts and other Swissness there, but, like all markets, you also find things that are mass produced - still, it is worth taking a look.
Here is the Spring promotion. Be sure to read through it to see when to purchase it, etc.
https://www.bls.ch/en/freizeit-und-ferien/ausfluege/fruehlings-aktion-thunersee
Edited to add: here is the link where you can purchase it online. The BLS site also provides a link to this site. I always have bought it online in the past, not in person.
https://shop.e-guma.ch/thunersee/en/gutscheine
