Lucas made some good suggestions for bases with the seasonal weather in mind - Luzern and Montreux.
<<"I'm aware about the weather being completely unpredictable during that time">>
That issue is of much greater concern in the high Alps, where you would go with the particular aim of seeing the magnificent vistas. Clouds and fog make that difficult, although there can be very nice days.
At the time of year, one thing definitely is predictable. In the high Alps, lot of hotels, restaurants and mountain transport ....although not all...will be shutdown because the skiing is no good, but the trails at high altitude hiking trails don't open up much before mid June.
However, mountain villages, such as Wengen, can be very nice to look a round in and some hotels remain open.
<<" We're not interested in trekking and would prefer doing sightseeing and spending more time in the local villages.">>
There are pleasant small villages all over Switzerland. Also, small cities with real old towns...not ersatz old towns rebuilt after WWII. Do you really want to spend time in villages? Many have nice inns. However, if you wish them to be a travel base, it is best to pick one on or near a train line that can get quickly to a larger city.
One forum visitor a while ago chose to stay in Merlischachen, with good access to Luzern:
Nice views from that village.
If you really want to visit smaller villages in the countryside, you have good access by train and bus, but in the countryside, the bus service tends to be once per hour, sometimes with a gap at lunch time. If you choose to visit the Emmental region (between Luzern and Bern) from a base in Luzern, many of those villages are on train lines and easy to reach....still perhaps only once per hour service unless you are at a junction point where a few lines come together. Public transport is high quality and covers most of the rural areas quite well. Buses ran into most small villages. Often, a short walk can be rewarding and fill time between buses or trains.
For maximum flexibility, not constrained to one-hour transport timing, renting a car can be helpful. You can get access to many scenic vistas that way. All of the attached images were taken from rural roads in the Emmental region. The one called the Lueg is so well know that there is actually bus service to it. The view of Trub and the Trachselwald panorama were not accessible by public transport; the other two were. The village of Trub, photographed from a high ridge near by, is itself on a bus line. Trub us nice to walk around in.
However, driving in the larger cities is sometime problematic, because the city streets follow old trails, not a grid, and parking a car overnight, usually free of charge in the villages or with a modest charge at rural train a can easily cost 55-60 CHF, for example, at the main train station in Luzern.
My solution has been to rent a car but stay in the countryside, and use both the car and public transport. Of course, that costs more.
If you want to stay in a village instead of the city...and Luzern is a really nice place to stay...old town, scenery, good restaurants, lake boats all year riund .. let us know, and we can make some suggestions.
Weather in the Spring -
My preferred time to visit is late May in the countryside, but not the Alps, and September, in the Alps as well, after the tourist rush in August. For various reasons, in 2014, I could not go in May, and went in mid-April. That year, April was unusually warm and dry ...almost no rain - for about 10 days in the middle of the month. Some rain here and there around the country, different regions can have widely different weather, but many sunny dry days. Ordinarily, there is a fair amount of rain in April and early May.
I was staying in Sumiswald at a very nice inn and drove up to a ridge overlooking the village. The Trachselwald panorama was taken on about the 15th of April. Sumiswald is at about 700 meters altitude. You can see that the deciduous trees are blossoming, but have not put out leaves. Lots of snow still in the Alps. N snow left in Sumiswald. The grass in the picture was greening up at the lowest place in the valley, but still not really greening up on the higher areas. By late April and early May, Spring would have put leaves on the trees, and the grass would be greener.
I have attached another picture of Sumiswald, taken on November 2, 2014.
Sumiswald is perhaps a bit larger than many of the villages, and not so picturesque, but makes a good base for my hiking and photography.