Understanding SBB

S, IC, RE or if it even matters.

It does not matter. These are just different classes of trains, but a ticket from A to B is good on any of them.
I do think your bad experience was a one-off.

The "+" shows how much it is running late in minutes. Phone app will help if there is a disturbance and you can get quickly rerouted if there is.

For your information "S" is an S-Bahn, basically a suburban commuter train. "IC" is an Intercity train, a high-end train connecting major cities in Switzerland, often with a restaurant car. "RE" is a Regional Express, a middle of the road train with no restaurant or snacks on board.

Since the S-Bahn makes a lot of stops and can take twice as long as an IC or RE train, stick to those, but a ticket is good on any of them.

Yes, as Sam says, it doesn't really matter, just an indication of how many stops the train makes.
S = S-Bahn = Suburban/commuter train. Stops at every stop
R = Regional. Rural train stopping at all stops.
IR = Inter-regio. Doesn't stop at every station.
IRE = Inter-regio Express. Even fewer stops.
IC = Intercity. Stops in big cities only.

They also have route numbers, for example S1, S2 etc. You will see S-Bahn maps for each city showing the routes. The route numbers are useful for regular travellers to learn, one-time tourists just look up your destination.
Some lines will only have an S or R train, stopping at all stops.
Main lines will have that, plus additionally IR or IRE or IC trains.

The important thing to know is that every route and every station has at least one train per hour, at the same minutes past each hour.

Remember: There will always be another train in 60 (or 30) minutes. So don't panic if you miss a train.
That announcement you heard probably said something like "train xxx is cancelled please take the train to yyy in 10 minutes, and change at yyy", which is why nobody was panicking and why they couldn't understand why you wanted a refund.

AND, if you want to get from Basel to Schaffhausen (or vice-versa), SBB is not the way to go. Both are in Switzerland, but because of the way the border runs, the shortest and quickest route is on a DB (German Railways) train via Germany.
If you go to the other station in Basel (Badischer Bahnhof, owned by DB) you can get a train direct to Schaffhausen, no changes.

Thanx guys. I'm going to cut and paste the codes into my travel guide.

Is the app worth downloading? We'll have cell service there so if it works I'll get it.

Yep, if the young man in customer service had just said that we'd have waited but he offered nothing so we were like ok, can we get a refund. Gotta work on my German skills this trip. I've used trams, trains, metro, etc all over the world and never been as freaked as that experience.

Actually we were at Badischer Bahnhof as it happened. It was just down the road at the Swissotel where we were staying. Loved that tourists in Basel got free public transport.

Of course the app is worth getting. You can buy your tickets directly from the app - no need for ticket machines. You can check schedules with the app - it's very useful.

We just came back from a week in Switzerland and I used the SBB mobile app multiple times a day to check schedules. We stayed in Muerren and had BO passes, so never needed tickets for anything, but the schedule information was invaluable. I found the app easy to use and a godsend when making last-minute transportation plans.

Thanx for all the help. My fears are easing.

The IC train goes once per hour between Zurich Hbf and Schaffhausen and is nonstop.
The RE train goes once per hour between Zurich Hbf and Schaffhausen and makes 2 stops along the way.
The S-Bahns all take lightly different routes making many stops. Two go once per hour and one goes twice per hour.
So all in all, there are 6 trains per hour.

Just chiming in to say that, I, too found the website confusing before I got to Switzerland, but once in the country, it was much easier. I didn't have cell service, but each night at the hotel on the wifi I would take screenshots of the platforms and times of the trains that we wanted. This worked well, with one exception in Lauterbrunnen when they had changed the track # between the time I took the screenshot and the time I got to the station. I was able to look at the signs at the station to find the correct track just fine. It will be OK, and the good news is, almost everyone in Switzerland speaks English so if you are truly lost or confused, you can usually find someone to help you.

The website is indeed confusing, especially when it comes to fares (it assumes you own a half fare card), but the app is a joy to use!