I'm talking about Bangkok metros with a s because there are in fact several existing lines and future lines which are oprated by 3 different companies: the BTS network which was historically Bangkok Skytrain, the MRT network, which was historically the underground network (Bangkok subway) and the Airport Rail Link metro network (ARL) and some suburban train lines operated by the State Railway of Thailand (see below for the details about each metro) . If all these networks are interconnected in several stations, they each have their own system of tickets or tokens. This implies that you have to buy a ticket or token each time you go from one network to the other one even if with the touchless payment by credit card, we gradually arrive at to a more practical global solution.
For tourists, it is mainly the historic lines of the BTS and especially the MRT which are interesting to get around Bangkok and visit the major attractions. First, the MRT blue line that stops in Chinatown district and goes to the Great Palace, Temple of the Emerald Buddha and Wat Pho (stop at Sanam Chai station). The Airport Rail Link allows tourists to easily connect downtown Bangkok and Suvarnabhumi International Airport while the new Red Line goes to Don Muang Airport. The Gold Line's 'only interest is to go the gigantic Hi-So Icon Siam shopping center.
Bangkok metros are connected to Chao Phraya Express river shuttle boats at BTS Saphan Taphan station (Sathorn station on the Chao Phraya Express) or to shuttleboats on the Klongs (canals): at MRT Phetchaburi station for the Khlong Saen Saep boats and Hua Lamphong for Khlong Phadung Krung Kasem boats.
In general, subways and skytrains, possibly combining them with a river shuttle boat system, are the best option to get around in Bangkok. It is affordable, you are in an air-conditioned environment, you are not stucked in the traffic jams, you do not breathe exhaust gases as aboard a tuk-tuk. Once you'll start use it, I'm sure it will quickly become your favorite way to go everywhere in Bangkok city. Especially since the network is developing like never before!
I invite you to download the smartphone application viabus for your research of public transport trips and itineraries in Bangkok. It exists on Android and iOS and brings together all the public transport networks to tell you all the possible routes according to your starting point.