Buses are the most common means of public transport. There are two tipes: with or without AC. They are cheap, a ticket for an AC bus being less than 20 bahts. There are a few things to keep in mind when using buses.
First, during rush hour you’re mostly sitting. You’ll see a lot of people sleeping in buses. That’s a good use of their time.
Second, it’s very hard to plan a trip using buses. When you take a bus, you have to tell the ticket person where you get off – the price of the ticket depends on the distance. That person does not know English. So telling them “To the Grand Palace” will not do. I’ve tried miming also, didn’t work. Best option is mobile internet + Google Maps, GMaps has bus routes and stop names.
Third, bus drivers are crazy. Crazy! We took a bus one evening. It was after rush hour. The bus was doing slalom between scooters, while also honking them for being slow. I was sitting down, clinging to bars with both hands, yet I was not feeling secure. Thai bus seats should come with seat belts.
A better way to get around is by subway. Bangkok has two types: BTS (also known as Skytrain) and MRT (underground). They are inexpensive (20-40 bahts), comfortable and fast. But… (you knew there would be a “but” here) they only go to some parts of the city (the more developed ones).
For a fast way to see the historical areas, I really recommend taking the Chao Phraya Express Boats. With just 15 bahts, you can get to the Grand Palace, Wat Arun or close to Khao San Road. The river is also a tourist destination by itself. You can see skyscrapers, luxury hotels and then wooden barracks built on stilts over the water.
The most touristy way to travel is by tuk-tuk. Tuk-tuk drivers know two things: English and that tourists have lots of money. So be prepared for hard bargaining or overpriced trips (or both). Also, watch out for tuk-tuk scams (google for examples). We took only one tul-tuk and the driver told us the trip would be almost free if he stops us don’t know where. We gave him a firm “no” (several times).
For travelling outside of Bangkok, you have minivans (cramp, not very comfortable and with little room for luggage), trains (we didn’t take any, but they seemed kinda slow) and planes (to destinations like Chang Mai or Phuket). For a flight to Phuket, round trip, booked a couple of months in advance, we paid around 4000 bahts (try Nok Air, Air Asia or Thai Air).