To go from Suvarnabhumi Airport to Khao San Road and vice versa, the most practical way is to take the airport bus S1 (service started on 1st June 2017)
There is one bus S1 every 30 minutes. You can take it at Gate number 7 on the first floor of passengers terminal Suvarnabhumi Airport. The bus will drop you in front of Wat Bovonniwet or on Chakrabongse Road near Soi Rambuttri entrance in Khao San Road or at Sanam Luang near Great Palace. YTese are also the palce where you take the same bus to go to Suvarnabhumi Airport
The trip takes arouns 1h30 (depending on the trafic) and cost 60 THB per person wherever you stop.
Note that there is also a free shuttle bus connecting Don Muang to Suvarnabhumi airport (see details here).
There is also a free shuttle bus service connecting the two Bangkok airports. All you need to use it is a plane ticket that took off from the other airport. The journey takes around 1 hour 30 minutes depending on the road traffic.
The Bangkok Skytrain is called BTS. The trains have air condition and let you go over the heavy road traffic in the capital but you can not go everywhere even if connected to the Chao Phraya Express baots, it can let you go to major temples in Bangkok.
Taking the bus to get around is clearly not the easiest mean of transportation to use for a foreign tourist in Bangkok neither the fastest one but it's the cheapest one. Bangkok bus network is wide and you can go almost anywhere but just need to be patient.
Only Thai city for the moment to have a metro (there is metro project in Phuket), Bangkok is the city in Thailand that offers more choices of public or individual transport (metro, bus, boats, tuktuks...). No, there are not just tuk-tuks and pink taxis to get around Bangkok !