I am learning the game using Google Translate on the German manual, which works somewhat poorly. Here's what I understand so far, fwiw. So far I only have tried the practice mode, not the networked mode.
To get started: click "Laufende Spiele" to see the available regions; click a region name and you will see the switchboards (Stellwerke) for that region. The list of switchboards also shows a difficulty rating (Grad) on the right ranging from 1=easy to 4=hard. Click the name of a switchboard; now you get a description of the switchboard (useless because the German is too difficult), a preview of the switchboard, and buttons at the bottom to start playing. For practice mode (Übungsumgebung) click one of the two buttons on the right showing two different times of day. (The two times may have different associated operating conditions listed in the upper right, such as traffic, weather, etc, again useless because the German is too difficult.) Now you get a white-background page with a box loading the Java game at the bottom, give it a minute to get to 100% and click "Starte Simulator" (finally, German words I can read!).
Crash course in getting a train to move when it's stopped at a signal: Look under "Zugfolge" at lower left (train sequence) and find a train whose number (left column) is red. That means the train is stuck at a signal---of course the eventual objective is to reduce or avoid trains getting stuck at signals, but the good thing is when you click the train, then look under "Fahrplan" (timetable) it will say "Halt vor Signal S151" (stopped for signal S151). Now you can easily figure out where the train is located, by looking on the switchboard for signal S151. Look again under Fahrplan to see its origin and destination (von Origin nach Destination) and the schedule of platforms it's scheduled to stop at (19:54-19:54 Zwt1). Now you're ready to send the train from signal S151 where it's stopped, to its next platform Zwt1. Find the platform Zwt1 on the switchboard, and create a route from S151 to Zwt1: click the red button on S151, then click the next signal the train should go to; the swichboard automatically sets the appropriate switches and then turns the signal to green. Continue in this way until you let the train get to Zwt1.
Getting trains to move before they get stuck at a signal: This will become obvious once you've done the above method. Click the trains proactively and check their timetables (Fahrplan) to figure out where they're coming from and to. Sometimes you can set the train's entire course to all green signals and then the train will take care of itself from there.
What if you click the signals in sequence but it doesn't turn the signal to green?: One problem is road crossings (Bahnubergang). After clicking the red buttons to try to set the route, it will make a dinging noise until the road crossing is clear; click the road crossing itself (outlined in a little yellow box) then click Bu at the top menu (in the black box). The road crossing now changes to allow the train to pass, then finally the signal turns green. Another problem is broken signals, which are a more advanced topic which I don't understand very well.
What if you set the wrong route?: If the train hasn't taken the route yet, you can unset the route by clicking the first signal, then clicking "FS auflosen" at the top. If the train has taken the route already, I don't know what to do, maybe just send it as close as you can to its intended destination...
What if you set a route that conflicts with one that's already set?: The switchboard is capable of remembering one next routing beyond the current one. If you do this, you'll see yellow blinking lights under both signals. After the current route is used, the new route will automatically be set. This can be very helpful to keep everything moving smoothly. If you want to cancel the next-routing, you can click the first signal, then "SP loschen" at the top, and the yellow blinking lights will stop.
Releasing trains from their origin: Some origins simply give you trains whether you agree or not. Other origins will make a dinging noise and flash a red light under a special origin button off the side of the track by the origin. You have to give the permission for the train to come in by clicking the special origin button and then "annehmen" at the top. On the other hand, you can actually deny permission by clicking the origin button and then "loschen" at the top, in which case the train will wait a while before asking again(? I'm not sure about this part).
Automatic signals: The only one I really understand is "Auto FS gr..". You can use this on signals that don't have any switches on their following track. Click the signal, then click "Auto FS gr.." Now the signal will stay green anytime the following track is clear, decreasing the number of clicks you have to make in general. It also shows a little white light to indicate it's in this mode. You can unset this by clicking the signal and then "FS auflosen". Another interesting one is "man. gl. FS": click it and then within 5 seconds click a sequence of two signals. Now, anytime you set a route to the first signal, the switchboard will automatically set a subsequent route from the first signal to the second signal.
Manual operation: In all the above methods, the switches are set automatically according to the selected route. If you want to set a switch manually, you can. Click the switch's tiny gray button, then click "Weiche". Once you have all the switches set the way you want to link two signals, you can click "UFGT" and then within 5 seconds click the two signals. If the switches are set correctly, this will set the route without changing the switches. Another manual operation is to allow a train to pass a red signal, for example if the signal is broken. Click the red signal, then click "ErsGT". The train will proceed excruciatingly slowly and it's up to you to make sure the switches are set the right way.
Ok, that's what I understand so far; I hope it's helpful to you if you're new to the game. Please post more if you learn any answers to the parts I don't understand yet