We would recommend spending at least 2 nights in Bangkok because there is so much to see and do. However, if you like big, bustling cities, you may want even more.
The cheapest and easiest form of transportation in Bangkok is Grab which is the Uber of SE Asia. Just download the app (iPhone | Android) and input your current location and your destination.
Chiang Mai is one of the most visited cities in Thailand due to the high concentration of temples in the very compact old town area. It is also a great place to visit an elephant sanctuary if you are so inclined.
Koh Tao is an island in the Gulf of Thailand that has some of the most beautiful beaches in the entire country. This little island offers some of the best scuba diving in Thailand, and is one of the best places in the world to get your PADI certification.
We would recommend spending 2-3 nights on Koh Samui. Opt for more time on Koh Tao as the beaches are more scenic and the vibe a bit more relaxed.