There's no airport on Ko Pha Ngan. The only way to get to the island is by boat
Most people fly to Ko Samui (airport code is USM) and take the boat from there, which is what we did as well. Ko Samui is another island just to the south of Ko Pha Ngan
From Ko Samui, there are multiple boat services to get to Ko Pha Ngan. There's no need to book anything in advance, it's much easier to deal with it once you land in Samui. We went on the day of the full moon party and was able to get a ticket after we landed
From my experience:
After you get your luggage at the Ko Samui airport (which is a tiny airport), you'll see a couple of booths advertising transportation services
Look for the booth that has "Ko Pha Ngan" or "Ko Pangan" written on it. Or just ask them if they go to Ko Pha Ngan. You'll find one in a few seconds
The company we found was called Seatran Discovery, which operated large public ferries between the islands. The cost was 400 baht per person, including taxi from the airport to the pier
After we bought our tickets at the booth at the airport with cash (they didn't take credit card), we basically just followed their directions and were ushered to the taxi, which took us to the pier (Bangrak Pier).
At the pier, we had to exchange our voucher with actual boat tickets. We almost didn't do this because they didn't give clear instructions at their booth at the airport. But it's easy enough we just gave the ticket office our voucher and they gave us the tickets.
The Seatran Discovery ferry we took left from the Bangrak Pier on Ko Samui, which was super close to the airport (10-min taxi ride). On the Ko Pha Ngan side, the pier is called the Thong Sala Pier, which is the main pier on Ko Pha Ngan
The ferry was pretty full the day we went, which was a full moon party day so the entire boat was filled with backpackers. The ferry ride was about 40 minutes
Buy your return boat ticket (Ko Pha Ngan back to Ko Samui) at the same time you buy your ticket to Ko Pha Ngan. That way you don't have to stress about securing a spot coming back
The Thong Sala Pier is smack in the middle of the Ko Pha Ngan island. From there there are lots of taxi drivers offering their services after you land. To the Haad Rin beach, which is on the southern tip of the island where the full moon party takes place, a shared red taxi (Songthaew) will be around 150 baht per person, and you'll have to wait until the taxi is full with around 8-12 people. A regular car taxi will be around 200-300 baht per person. Unfortunately there's no Grab (the Uber of Southeast Asia) on the island
Try to land in Ko Samui by 3pm. The last big ferry departs at 5:30pm from Ko Samui. After that, you'll have to get to Ko Pha Ngan on a speed boat, which is much more expensive than big ferry (800 baht was the price we were quoted for a speed boat)
This is the schedule of the Seatran Discovery company from Ko Samui
There's another major ferry company called Lomprayah that competes with Seatran Discovery. Their pier on Ko Samui is much farther from the airport than Seatran, but their ferry price is about the same. I didn't see their booth at the airport, but their website is: https://lomprayah.com