Train tickets, even departing from Krung Thep Aphiwat Central Terminal Station, can be purchased at any train station in Thailand, including Hua Lamphong Station. In Krung Thep Aphiwat station itself, there are ticket-selling boots on the left when coming up from the MRT metro, near gate 14, or opposite gate number 3, the one where the shuttle bus and most taxis drop you off. Train tickets can also be purchased on the internet on the SRT official website dticket.railway.co.th or on the 12go website for the most common journeys.
Trains of the Northern line (to Chiang Mai) and the North-Eastern line (to Nong Khai or Ubon Ratchathani) leave from platforms 1 and 2 and arrive at platforms 5 and 6. Trains for the South of Thailand depart from platforms 7 and 8 and arrive at platforms 11 and 12.
The platforms are located upstairs. Access to them is only open 20 minutes before the departure of the train. It's a bit like boarding a plane at an airport with waiting areas with seats and a security cordon blocking access to the platforms. Then, there is an announcement saying that boarding will be open, everyone queues, and shows their tickets to a SRT officer before they can access the escalators, get on the platform, and board their train. But no need to show your passport.
Near gate 4, there is a small restaurant area called Food Station. There are also a few snack and drink vending machines in the station as well as spaces to charge your electronic devices. There is a LockBox left luggage service.
There is also since the commissioning of the new station, a free shuttle bus system that connects Hua Lamphong station to Krung Thep Aphiwat. It can be picked up outside Hua Lamphong station, near the metro exit, and drops you off in front of Krung Thep Aphiwat station, gate number 3. In the other direction, the shuttle bus can be picked up at gate 12 at Krung Thep Aphiwat Central Station. The journey takes 15-20 minutes with an average bus every 15 minutes from 04:30 to 23:00 but at some hours it only goes every half or even every hour (see timetable below).
In theory, you need a train ticket to get on the bus, but the drivers don't seem to ask for it. have a look at the latest known timetable for the free buses between Hua Lamphong and Krung Thep Aphiwat Central Terminal Station. At the top, this is from Krung Thep Aphiwat to Hua Lamphong, and at the bottom buses departure times in the other direction, but they are the same times.
The MRT metro and shuttle buses are the public transport to connect downtown Bangkok and Krung Thep Aphiwat station, but there is also a station of the Red Line trains, which conveniently goes to Don Muang airport, in Krung Thep Aphiwat Central Terminal Station. It is taken at platforms 3-4 to go to Rangsit and platforms 9-10 to go toTaling Chan. Very practical for connecting a domestic flight and a train or vice versa. To get to Suvarnabhumi Airport, it will be a little bit more complicated since you will have to take the MRT metro in the basement to Phetchaburi station and then change to get on the Airport Rail Link.
Several regular bus lines also stop at Krung Thep Aphiwat Central Terminal Station, some of them connect it to the Northern Bangkok Bus Terminal (Mo Chit Bus Terminal), opposite which is the Chatuchak Minibus Station. Their stop is located outside between gates 10 and 12.
Taxis drop off passengers at gates 3 and 4 and pick them up at gate 7. Due to the scarcity of customers, it is still very complicated to force them to start their taximeter from Krungthep Aphiwat station.
For more detailed and accurate information, I invite you to download Richard Barrow's Krung Thep Aphiwat Central Terminal Station comprehensive guide.