There are now 2 daily direct international bus services from Bangkok to Siem Reap. For some strange reason, I recently saw one of these buses parked at a bus station in Phnom Penh and on the same day, another Siem Reap-Bangkok-Phnom Penh international bus was parked at Koh Kong, both well away from the routes they should be serving. Anyway, I digress.
It's easy to travel on the international bus from Bangkok. It would be best to show up at Mo Chit bus station run by the "Transport Company", where the bus departs and buy your ticket the night before or if you can't manage that, very early the next day around 6 or 7 in the morning at the latest as I believe the buses depart around 8 and 8.30am or 9am. Thai and Cambodian buses run on an alternate day basis - one day it's a RHD Thai bus, the next it's a LHD Cambodian bus run by Nattakan Bus company. Don't believe anyone who claims there are any other cross-border services, or even no cross border services. These two companies, that operate in tandem are the only ones offering a cross border service.
Once at the border everyone gets off, sorts out their visa (note that it isn't difficult to pay the proper $20 for a tourist visa or $25 for a business visa at this border despite what some people claim - just insist, be firm and friendly and it will take less than 5 mins - I speak from personal experience having obtained a visa there twice, one tourist and one business and both times I didn't pay extra except the second time an extra $2 (bargained down from $3) for not having a photograph, but that was my fault) and then you reboard the same bus after Cambodian immigration. The bus company will give you a sticker to identify you for reboarding purposes once you have crossed the border. I believe they will tell you what time the bus will depart from the roundabout on the Cambodian side, probably around 1 hour later, which should be enough time to get through. Still, due to occasional crowds it could take longer and consequently, there may be some pressure exerted by the bus company to speed things up and one way they claim they can do this is by offering to do your Cambodian visa for you for a "fee". Don't let them, since they will charge more and it won't be any quicker than doing it yourself.
I have heard that even though it's a government bus, touts may try to steer you towards some kind of accommodation in Siem Reap that provides them a commission (this doesn't happen in the opposite direction though), so insist you have your own accommodation and don't let the drivers take you anywhere you don't want to go.
If you go down the DYI route (meaning you catch one form of transportation to the Thai border, walk across and then change to a different form of transportation on the Cambodian side) then you'll be under absolutely no time pressure at all, especially since it's possible to get to Siem Reap in no more than 2 hours from the border and now that the border closes at 10pm, you should find yourself with plenty of transport options once across the border.
The best way of reaching the border from Bangkok is via a minivan that leaves from near the Victory Monument BTS station for 230 Baht. These vans take a mere 3 or so hours (although often around 4 hours in the other direction due to traffic) to reach the border, by immediately getting on the expressway locating right in front of the van station. Departures are approximately hourly starting as early as 4am. You want to go to "Talad Rong Kluea" or ตลาดโรงเกลือ in Thai, which means "Salt Factory Market" in Thai, and refers to the market right at the border crossing. Buses and trains stop in Aranyaprathet, rather than the border. Heading only to Aranyaprathet, some 6km away from the border is inconvenient and invites further costs and rip-offs as you'll have to negotiate for a tuk-tuk, motorcycle taxi etc. to the border whereas the van takes you all the way there.
Once you get off the van, you walk across the border, do all the formalties yourself including obtaining a visa on arrival. Note that as soon as you leave Thai immigration and enter Cambodian territory move to the RIGHT and avoid the scam on the LEFT with the fake health forms. The visa-on-arrival counter is located about halfway between the Thai and Cambodian immigration posts. Obtain your visa there, then keep walking on the right past a couple of casinos to Cambodian immigration.
Once you are past there, buses and taxis head for Siem Reap. I'd actually recommend a taxi, which if you are sharing it with a few other travelers is no more expensive than the bus. If you need a bus, make sure you go with an official bus company and book tickets at the bus station by getting there yourself, rather than paying a tout who will inevitably take a commission for himself.
Never catch the Khao San Road scam bus - apart from being more uncomfortable and costing more than the official government bus due to all the added extra "costs", you will find it difficult to avoid the visa and accommodation scams and you won't even be on the same bus all the way. Only first time backpackers make that mistake - don't be one of them!