The SkyTrain is the fantastic mass transit train that goes throughout the city.
Nothing in America comes close to this bad boy. It's fast, it's clean, it has some really cold air conditioning, it's... slightly complicated. But, it's worth the time that it will take for you to figure it all out. Check the promotions, (SkyTrain) it's fairly cheap to travel throughout the city and they do have specials/promotions every now and then that can save you even more money. The cost is kept low for Thai people! It could be as low as 22 Baht per trip. At today's exchange rate that is something like 67 cents each way per person. If you're staying somewhere on Sukhumvit, you should be able to go nearly everywhere that you want to go, without using a taxi, just take the SkyTrain. Worst case scenario, you'll take the SkyTrain to the Subway, then take it where you'll need to go. There are places you cannot go on the SkyTrain, (places north of the city) then, you'll need to take a tuk tuk or taxi to your destination. (which is fairly cheap)
The Bangkok SkyTrain is awesome, it's a no muss, no fuss, fast, cheap, air conditioned mass transit!
And now you can take a train from the airport! (see here) or below.

The Commuter Train will make eight stops throughout the city, taking 30 minutes to travel the 28 kilometers (17 miles) to the airport and linking with both the *Subway and **Skytrain systems in Bangkok.

* Subway Link:
Phetchaburi station for MRT (subway) Blue line
** Sky Train:
Phayathai Station for BTS Sukhumvit line.