Thailand has a very efficient and cheap travel infrastructure, provided you understand how it works
There are numerous internal flights at moderate prices, but things change too fast for it to be worth trying to list anything here. Search the web, or ask a travel agent in Thailand. It's mostly cheaper to book flights locally, rather than trying to book them from your home country.
Trains tend to be reliable and frequent but slow, and you may need to book a day or two ahead on popular routes at busy times. A good way to see the country if you're not in a hurry.
Thailand has good roads, but it's a big country, so journeys can take a long time, and driving standards may not be what you are accustomed to. Public buses (very cheap) and private ones (fairly cheap) operate on most major routes. Bus stations generally have a separate ticket counter for each route, which eases the language problem, and public buses usually have the destination painted on the side in English and Thai.
The best way to travel by road in comfort or with luggage, though you may spend more time staring at traffic jams than moving. Fares start at B35 and increase by a few baht per kilometre, or per minute if stationary. Make sure the driver turns the meter on and, if you know the route, that its indicated distance is correct!
Once upon a time, no taxis had meters. A few non-meter taxis still survive and haunt tourist hotels. Fares must be negotiated before you set off, so unless you know the route and the scale of fares you are likely to be overcharged. Not recommended.
The ubiquitous three-wheeled motor rickshaw, sometimes also called a samlor (literally, "three wheels"). Again, fares must be negotiated beforehand. Do it once for the experience if you must, but usually they are no cheaper than a taxi, less comfortable, and you're exposed to the heat, dust and fumes.
Every street corner has a group of motorcycle taxis; the riders wear numbered uniform vests. Fares mut be negotiated beforehand. Very cheap, but not for the faint-hearted. But in an emergency the "Bangkok helicopter" may be the only way through the jams.
Bangkok buses go everywhere, but you need a detailed and up-to-date bus map to make sense of them. Very cheap. The air-conditioned ones cost slightly more, but that's still very little.
The best way to stay cool and sane in Bangkok is to arrange your journey so that as much of it as possible is off the road, just using a minimum-fare taxi ride at the beginning and end. Fortunately there are a number of alternatives to the roads, and transport on some routes has become much easier over the last few years.
These boats are essentially a bus service, right down to the low fares and the style of the conductor's ticket roll, stopping at piers up and down the river throughout daylight hours. See this map of the route. This is an excellent way to see the city and escape the gridlock
Water taxi services also operate on some of the other canals (khlongs.) By the time you've found your way around well enough to use them, you won't need my advice.
The Skytrain (Bangkok Mass Transit System, BTS) provides rapid air-conditioned elevated train transport along two routes in the city, for fares typically between 15 and 30 baht. For single journeys, you buy a ticket from machines in the station. These machines only take coins, but the ticket office can change notes. There are also various kinds of multiple-journey ticket.
See this page for routes and fares.
The latest arrival, in 2004, is the Metro (Mass Rapid Transport Authority, MRTA) underground system, with a route which fills in some of the gaps in the BTS. Some stations are designated as interchanges with the Skytrain, but there's no integration: the "interchange" entails a walkof a few hundred metres to the corresponding station. Ticketing options are similar to the Skytrain, but the machines take notes as well as coins. There is no cross-ticketing between the systems.
See this page for the route and fares.
In most towns, there are no taxis or scheduled buses. In some places there are tuk-tuks, or even pedal-powered samlors, which work just as in Bangkok - you negotiate a fare before setting off. However, the most usual means of transport is the songthaew (literally, "two rows"), a pick-up truck with two benches installed in the back. These function as both bus and taxi: some have regular routes, others do not. There is often a colour scheme (different from town to town) which shows which they are.
Some songthaews ply regular routes, either in town or between villages, and function just like a bus - you flag them down, ask if they pass your destination and (if so) hop in. When you want to get off, you ring the bell, and then pay the driver the fixed fare. These fixed-route songthaews often display the route, but only in Thai.
Others are more like a taxi - if you flag down an empty songthaew, you can charter it to go anywhere you want, but the fare will be relatively high. The catch is that, once chartered, the driver can still pick up other passengers if their destination is on or near your route, and charge them a lower "bus" fare. So, unless your destination is somewhere out of the ordinary, it will be cheaper to flag down one which already has passengers, and hope it's going your way.
Many places offer motorcycle, bicycle or car hire, at modest rates. If you're staying anywhere for a few days, one of these may be the best way of getting around. Be sure you're well insured, and judge for yourself whether you can cope with the traffic. Outside the congestion of the few big cities, driving on Thailand's main road network is no problem for the experienced driver, but it will test your defensive-driving skills, as there are vehicles on the roads with a wide range of speeds and random lane changes without prior warning are the norm. In remoter areas, paved roads may abruptly become steep and bumpy dirt roads which are only recommended if you have 4-wheel drive.