The MRT subway is a convenient way to get around. The fares start at 16 baht for a single journey.
Take the stairs from street level to the station. Upon entering you have to go through a metal detector like at airports.
Go to a ticketing machine. You will find there a touch screen with map that displays, how much the fare is when you select your journey. After you payed your fare you will receive a small plastic coin. This is your ticket.
Most ticket vending machines accept coins and also bank notes. Day passes are also available. You can purchase them only at the ticket counters located at some stations. Go through the fare gate by placing your coin/token on the scanner.
The system works similar to public transport in Europe, look for a sign that tells the name of last station of the line. This will show you in which direction which train will take you. Board the train. On the train they will make announcements about the next stop both in Thai and English. Get off at your designated stop and exit the station through the fare gate.
There’s currently 1 line running through the city. The government of Bangkok is still working on extending the existing network.
Here’s a video by DiscoveryThailand, which explains the system pretty detailed.