To go to Don Muang Airport from Bangkok and vice versa, besides taking the train from Bangkok Central Station (Hua Lamphong), the cheapest solution but also the slower, there are 4 bus lines called ''Airport Express'' connectiong travellers to Bangkok Skytrain BTS or Bangkok subway MRT, Bangkok Northern Bus Terminal (Mo Chit), Lumphini Park or Khao San Road. These buses are ate located at exits number 7 and 8 in Terminal 1, just go out at those exits and you will see them.
There is the classic A1 route:
There is the A1 route that goes only to the metro stations BTS (Mo Chit) and MRT (Chatuchak), the most frequent one. It takes 30 to 45 minutes to reach the Mo Chit BTS station (or MRT Chatuchak).
And from 5th December 2019 and the opening of new BTS stations closer to Don Muang airport, every 20 minutes there is a new A1 route that goes to the following stops:
To come to Don Muang airport from Bangkok there is a bus stop a few hundred meters after BTS station Mo Chit or MRT station Chatuchak (same side Chatuchak park).
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 !