We plan to be in Berlin several days from Saturday through Monday, using public transportation on all days and visiting the usual tourist spots. Is the purchase of the Welome Card a good value? How do you use it on the S and U-Bahns?
Whether the card is a good value depends on what sights you want to see in the city and whether the card covers them. Unless something has changed since 2015 (entirely possible):
The plain Welcome Card doesn't include any of the Museumsinseln museums. In 2015 there was a Museumsinseln add-on you could buy, but that turned out to be highly problematic (read on).
The Welcome Card with Museumsinseln add-on didn't allow me skip ticket lines on Museumsinseln. Those can be an hour long or more.
When one of the Museumsinseln museums had a special exhibition with an extra fee, I was charged the full entry fee (basic entry plus special exhibition), receiving no benefit from having the Welcome Card with add-on.
Keep in mind that Berlin is a very large city. There are widely-scattered sights. You will spend a fair amount of time traveling between those sights. That is time you will not be inside a sight. You can easily buy a day-pass covering Berlin transportation; there may well be a 3-day pass also.
There are some free sights in Berlin that you may wish to see. They would chew up time, leaving less time for the sights covered by the Welcome Card.
So there is truly not a one-size-fits-all answer.
How do you use it on the S and U-Bahns?
Answeed in FAQ of Welcome Card website.
Welcome Card was a huge waste of money for us. We only used it for one discount (Olympic Stadium) and would have been better off buying daily passes for public transportation. We had to buy the Museum Island pass separately, no welcome card discounts. Look closely at what redeemable offers it has before purchasing.
Just a general comment: I love a bargain. I travel a lot in Europe. I have analyzed quite a lot of city cards to see whether it appears they'll save me money, given that I spend a lot of time in museums (often costly, but time-consuming so I don't go to a lot of places on a single day). I've never encountered a big-city card that was clearly going to save me a significant amount of money. The best case has been that I thought the card might generate a modest savings. In the real world, that savings don't always materialize. On balance, I don't like having a card in a big city. It tends to make me alter my preferred approach to tourism, which is to wander from place to place, absorbing what is around me. If I see an intriguing side-street, I want to be free to walk down it rather than thinking, "The sightseeing card is about to expire."
I certainly don't want to skip an interesting free sight because it isn't covered by the card. Nor do I want to contort my city itinerary to be sure I go to widely scattered covered sights on the days a card is valid if it means I need to revisit some of the same neighborhoods after the card expires to see the nearby places that are free or not covered.
Don't get too mesmerized by the local-travel component of a city card. Most cities (including Berlin as I mentioned above) sell transit passes separately.
I have found sightseeing cards offered by smaller cities (including Orvieto and Padua) to be very worthwhile, because I was spending enough time in those cities to take advantage of most of the covered sights, and they were not individually very time-consuming.