You may have noticed a giant tower looming over the town:
This is part of the train station. The main station is the checkered and unlabeled building across the street from the market:
Entering this building you will be beset with many many buttons and switches. We will cover the general design of the train system.
The Train Station in town is the primary hub. Treat each direction (N, E, S, W) as having their own independent addresses. An "address" is an 8 bit number, expressed in binary, such as "0000 0000". Each cardinal direction can share addresses, so all four directions have an address "0000 0000", "0000 0001", and so on.
Each cardinal direction currently has another station at the end of it, called a substation. North's first substation is "0000". Currently there are no further substation, but the next one down North's line would be "0001" etc. This is your "Primary Signal" as shown in this screenshot:
For now you will always leave this as "0000". Setting this to a number higher tells the station that you want to "go past" that substation to the next one. Because no other substation exists, you will simply fly off the tracks.
Each substation has public outlets with an "address":
These are public outlets for players to connect their own railways to. These are always OUT tracks. If you try to ride a minecart into them you will get put into a special holding area. You can build a rail that splits off and connects to the "ENTRY" track in order enter the station and either stay at that station or continue back to town.
Each outlet has a unique address as shown in the screen show above. The final four numbers are the important part of the address. The last four numbers are the Secondary Signal:
The secondary signal tells the stations which exit point to shoot you out. If you choose "0000" you will instead stop in the train lobby.
Launching a minecart:
Choose your cardinal direction. These directions correspond with the direction you can get from the ingame information (F3) or from the map renderer's compass (top right corner).
If you have a specific address (there probably aren't any at the moment), you'll want to select it on the Primary and Secondary wall. (Up is 0, Down is 1). Otherwise make sure all switches are up.
Once you have the correct direction and address you'll want to go over to the launchpad and set your cart down on the angled powered rail (that will be in the off state).
Then hop in the cart and click the "go" button:
Wait for the launcher to launch you. This will be important later because the train station is actually doing some mathy redstone stuff in the background during that short delay. Once you get to the substation there will be a couple of delays where your cart stops. This is normal! The station is decoding your signal. Do not get out of your cart; your trip will resume shortly.
Important: The island is large and the ocean around it is even larger. You will be in for quite a long ride, so sit back, enjoy, and maybe go afk. Depending on which section you are riding down you may be in for a 5+ minute ride.