I really enjoyed the variety of public transportation in Bangkok!! I especially liked the cashiers on the buses and boat taxis that would come around and take your money. They were especially helpful in letting me know where to get off - although many of them did not know English (understandably), but bi-lingual maps and some gestures go a long way!!
Traveling by bus is probably not the easiest thing to do as a newcomer to the city. However, if you are traveling with someone who lives in Bangkok or can find a good guide, they are cheap and convenient. The buses without air-conditioning are cheaper. However, the air-conditioned buses are much more comfortable than the non-air-conditioned buses.
Taxis are plentiful and cheap in Bangkok. Again, Thai language skills help, but with a good bi-lingual map or cards with Thai phrases on them, you can make do.
There are also water taxis, which function much like a bus, but on water. All along the banks of the river are docks for water taxis. Some simply do circles, taking people from one bank to the other, while others ply the length of the river. It gives an interesting view of the city from the river. There are also some that are available for individual hire or that will do tours throughout the canals.
Tuk-tuks are three-wheeled contraptions with the front of a motorcycle and a two-person seat in the back. In many ways, these are more for fun, but sometimes they can get through tight spots where a taxi couldn't. If you want to take a ride, though, bargain for the fare in advance.
Finally, there are motorcycle taxis, which I did not have the joy of experiencing. However, sometimes they are indispensable if you are visiting place in a narrow soi (alley).
With so many ways to get around, exploring Bangkok couldn't be easier!