Since you are already on Ko Tao, chances are you also took a Lomprayah ferry to get here, so you already know where the pier is. If not, it is easy to find. Mae Haad harbour is very small and you can ask anybody. It's in the small street that runs parallel to the harbour, below the central 7/11.
If you booked online, all you need to pick up your ticket is your booking number as given in the confirmation email. You can pick up your ticket a day early, to save you the hassle in the morning, because there will be a lot of people lining up for different boats. The middle counter (number 2) provides services for destination Bangkok. If you are lining up outside, check what people are lining up for. Luckily, I was able to sneak by, because everyone else was going to a different island.
That is also the reason why you should pay attention to the color-coded stickers: lots of people, lots of destinations. Put the round one in a well visible place on your T-shirt, so the service staff can help you in case you are boarding the wrong boat, and the long one on your luggage, so it will go to the same place as you. Make sure to tie that last one around a handle, it doesn't stick well. Staff will check and rip off your ticket, and you will need to keep one half of it for later.
Then you just need to wait for the ferry to arrive. Several boats are leaving from that pier in short intervals, so again, make sure to double check you are lining up for the right one. Once the ship is tied up at the pier, the freshly arrived boatload of tourists has to depart first, so allow some time for that (20 or so minutes should do). At this point the staff can get a bit hurried and can become impatient when people are slow or in the way, don't take it personally, they are very efficient.
The ferry has three seating areas: Downstairs (inside) with the most comfortable seats, in the middle (outside with sun roof) and on the top (outside without cover), both outside areas have metal benches. If you are prone to motion sickness like me, a bit of fresh air will be the better choice. Stay in the middle of the boat, where it does not move that much and keep your eyes on a point far away at the horizon. Toilets are located on the bottom floor in the back of the boat and look like ship toilets look, with a bit of water splashed everywhere, but otherwise clean. Because the tide was low, we were all asked to assemble in the front of the boat, (I'm assuming) so that the ship's propellers could move freely and get us out of the bay.
The journey takes about 2 hours and can get rough, if the weather is really bad. In that case they will close the outside areas and you have to stay inside. After you arrive, just take your luggage and walk down the pier towards the office building. At a booth to the left from where you exit the office building you can exchange the remaining half of ticket for a bus and seat number and another sticker. Make sure to have your numbers ready when the bus departs, they are strict about who goes where.
Then you will have about half an hour before the bus departs. There is a mini mart to the opposite side, if you need anything while waiting, and next to the booth is a food court with Thai food to the left and Western food to the right. There is also an array of clean toilets.
The bus had only regular reclining seats, so it is spacious but not exactly a sleeper bus, and the 8 hour drive may get a bit uncomfortable after a while. If you plan on sleeping, a travel pillow or a folded sarong helps a bit, and so do earplugs. The bus is air-conditioned and has a toilet. I'm only 1.60 m, so I had plenty of leg space, but anyone taller than 1.90 m should expect some discomfort and probably chose the aisle seat over the window. At 16:00 there was a 30 minute break at a market with snack food. The left counter has decent Thai food for about 100 Baht per plate, including some vegetarian fried rice and tofu, and the left offers Western food such as hot dogs. There was also a huge bathroom with clean toilets, but remember to bring your own toilet paper and flush properly with the bucket provided.
Generally, the bus company is exceptionally punctual and I was never late for anything when I traveled with them. So at 20:30, right on time, we arrived at the Lomprayah bus station in Bangkok, near Khao San Road, where a bunch of pushy taxi drivers will be waiting for you, but a firm and polite “No, I'll walk” will do the trick. Enjoy your time in Bangkok!