This is pretty easy, once you have at least 25 pounds; that'll get you 15 train trips, which is enough to get you started.

Turn left when you enter the train station and click on the monitor to the left of the closed ticket window. Just click on the number of tickets you want to buy, and they'll be dispensed below.

Each ticket gives you a roundtrip -- that is, out to your destination and back to the station again uses up one ticket. You'll need 25 to 30 tickets in the course of the game because you'll be popping back and forth between the different locations many times. Tickets are cheaper if you buy 15 at a time, but you can buy them 5 at a time if you don't have a lot of cash.