Thailand is a big country, larger than the UK.
Where in Thailand will you be visiting?
"What's the cheapest way to train and roughly how much?"
What does this mean? Training to be a Thai food chef? Training to be a masseur? Training to be a boxer or just want to ride trains?
Different cities have their own variety of local transportation. Bangkok is not that expensive for city transportation. There is good rail service. (Skytrain, Airport Rail link and a subway. They are connected.) It is possible for to use cheap rail service from Suvarnanabhumi Airport to get to many hotels in the city area for a low price. But not all hotels in Bangkok are close to city rail service. In Bangkok you ride a tuk-tuk for fun but not for regular transportation. Meter taxis are cheaper when you use the meter fare and do not pay a flat rate! Tuk-tuks in Bangkok often come up in scams to avoid! Do a Google search for scams in Bangkok.
Other cities in Thailand can be reached by bus, train and by flying. There are some scams involving buses booked from Khao San Road travel agencies. Best to use a real bus station for long bus rides even if you must pay a few cents more. On buses, trains, planes even in taxis DO NOT leave any passport, cash, jewelry, computer in any checked in bag. On the bus keep money and passport on your body and place your bag in a location where you can easily see if from your seat. Have bags you can lock!
What kind of training? Google search can be helpful to narrow down some cities.
Come back with some specific questions and locations.