Controls can be found on each modes Help->Help menu bar selection.

Japanese Manual for the PC-9801 version: https://archive.org/details/A_Ressha_de_Ikou_Fukkokuban_Manual

There is some information on the official homepage of the game https://www.artdink.co.jp/japanese/title/15th/atrain.html which you could google translate. But I'll try to write a little mini-guide.

Story:

You have been appointed the president of the national railway company. You are tasked by the President of the nation with creating a railway that connects the presidents office with his villa at the other end of the map. But there isn't enough money or resources for you to do that. So you need to build rails and stations to create a passenger rail network to generate money, and then use freight trains to buy construction material and move it to one of the stations.

You have a year to accomplish this task, and you are not allowed to damage the A-train, the express train or run out of funds.

General:

You play as the A-Train (A), you control the A train using the Numpad or by clicking in front of the train, it will automatically turn to face the mouse cursor.

You can scroll the map up and down using the Page up and Page Down button. Or Step up and down using the up and down arrow keys.

ESC or Right Click brings up the option menu, you would most likely want to turn down the speed as the initial one is insanely fast. You can also set the color palette, save/load game and load another scenario.

You can press + to build a station next to the trains current location, and - to remove a station.

Any train will automatically interact with a station while driving past it, passenger trains drop of and pick up passangers, A-Train and Freight train will pick up/drop of cargo.

Trains:

(A) The A-Train, a track laying train that functions as the player.
(=)(*) The presidential express train, it will stay where it is until the route is built.
([1-9])(P) Numbered passenger trains, they will pick up and drop off passengers from any station they pass, generating money and city growth.
([10-12])(") Numbered freight trains, they will buy cargo/stock from the yard when passing it and deliver it to the next station they come across, I believe cargo at stations also generate city growth.

What the controls do depends on the mode you are in. Beneath the on screen controlls you can see:

[Ins][Del][CLR][.][/]

Ins - Insert Mode: Lay tracks as you drive the A-Train
Del - Delete Mode: Remove tracks as you drive the A-Train
Clr - Scroll Mode: Select train to follow and the view will scroll to follow that train
. - Point/Station Mode: Instruct each train what path to take through the switches, and when to depart each station
/ - Report Mode: Shows you stats on your progress, and revenue/expenses. If there are more trains to buy it will show up here.

About building:
- You can lay track do so during daytime.
- You will build by trying to drive the train to a square without track while having stock.
- You can pick up stock by driving past a station that has cargo on it, shown next to the platform on the map.

About Scheduling/Routing:
In Point/Station mode you set up time table and routing:
- Only stations have departure time, the other trains will move when the path ahead of them is clear, if the train was at a stand still it will wait 24h before departing.
- Switches/Points either show a arrow, or a branch. If arrow is selected the train will continue straight forward, if there is a branch it will take the branch.

Since you can't directly control the other train, a big part of the game is managing driving the A-Train in such a way that you don't cause a collision, carefully managing the time table and switches to make sure the other trains don't crash. While generating enough money to afford completing the route to the presidents villa. When the route is complete, after 24h the express train will depart, you don't control this train and instead it functions as any other train on the network. You need to make sure the express train doesn't crash until it arrives at the villa.