There are a plethora of transportation options in Bangkok. It's a big city, so your most obvious option (walking) won't get you too far. Especially as you're likely to be walking beside a major roadway full of stationary diesel buses and tuk-tuks in the Bangkok heat, which happens all too frequently. How about getting on one of those buses? The air-con ones cost more (although they're all cheap - almost every transportation option charges by distance, buses tend to be in the $0.30US range), and all of them are subject to the whims of traffic congestion. The biggest trick, though, is figuring out which buses go where. It's pretty much the cheapest way to get around town other than walking, but there's nothing a North American would think of as a bus map. The best you'll do is a map with numbers alongside each road. Look at those numbers, look at your destination, and hope you can match numbers. Even then there are probably unlisted routes - I've ridden those at the direction of locals and arrived fine even though the bus isn't mentioned at origin or destination on the map. There are several more varieties of buses that I haven't got the hang of yet, most notably mini-buses.
The best way to travel here is the Skytrain. I'm lucky to be staying at a guest house a block from a Skytrain station. Fares vary by distance from about 15 to 40 Baht (that's about a dollar maximum) and you ride a new aircon train above the traffic. Unfortunately the system is fairly new and covers only a relatively small portion of the city. Your second best choice (which links up with the Skytrain in at least one place) is to catch public transit boats running up and down Mae Nam Chao Phraya. While you're obviously limited to where the river runs, a lot of tourist destinations are on the river because the city grew up around it. Costs on the river are comparable to buses and Skytrain.
If you want to get away from the public transit boat stations, you can pay for the services of a long-tail boat. I have to admit I'm fascinated by these ... Take a long, skinny, shallow boat. Mount a big gimbal at the back of the boat. Make it strong, because it's supporting a four, six, or eight cylinder car engine complete with transmission about two feet above the deck. Run a six meter (20 foot) shaft off the back of the boat with a propeller on it. Run a two meter (six foot) bar forward to the driver, and equip it with a couple odd levers for throttle and shifting gears. Then de-tune the engine so it belches black smoke whenever it's in use (or start with a 25 year old engine in the first place). Now crank it up, and send it skidding around the river at 30+ miles per hour, with the driver leaning about, swinging the entire motor and transmission to direct the boat ... Now you can charge 500 Baht per hour. I'll try to get a picture at some point.
At the top of my shit list are the tuk-tuks. These are three-wheeled motorized open vehicles that are ubiquitous in the city and across much of India (probably lots of other places, these are the ones I know). I'm pissed at them because they spew fumes, and because their drivers will cut across three lanes of traffic just to yell "where you going sir?" at any farang. Near the Grand Palace, you'll average ten offers per block, and they can be persistent. They also cost about the same as a taxi, although the taxi has better seats and aircon. A big fare on either is 300 Baht, about $7.50US.
If you're really brave, there are motorcycle taxis. These have much more manoueverability than eiter taxis or tuk-tuks, but a relatively low life expectancy as they drive you between stationary cars on heavily trafficked roads.
Scooters and motorcycles are rampant in the city, although the largest (legal - the law gets bent occasionally) displacement is something like 150cc, smaller than anything a North American would even recognize as a motorcycle these days. All of them lane-split (travel all around stopped cars) so every long light has a horde of bikes at the front of traffic.
I've read about and seen a couple samlors. These are essentially a pick-up truck with bench seats in the bed. They're apparently pretty common outside Bangkok - you wave them down like a taxi, but the cost and behaviour (they tend to stick to a specific route) is more like a bus.
One option I haven't tried is the subway, and I'm sure there are at least a couple more options I haven't mentioned here.
And finally, a joke. Unfortunately, it requires more explanation because I know that most of my friends will never have watched the TV show the joke references ... MTV has a half hour show called "Pimp My Ride," hosted by the fallen rapper Xzibit. People with really terrible cars send them videos begging them to "pimp my ride." The lucky individual chosen for the weekly show has his car picked up by Xzibit, and the car is then "pimped" (usually in an exceedingly bizarre way) by a California custom shop. It's worth watching precisely once: the shop is inventive in the things they do to the cars, and it's funny the first time.
So, a few days ago I was walking down Khaosan road, and I saw a guy wearing a shirt that said "Pimp My Tuk-Tuk." Okay, I thought it was funny ...