The headset. It doesn't do so much. It connects the fork's steerer tube to the head tube, allowing you to steer the bike. And that's it. So how do you decide what headset to get? I don't have much experience with them but the overwhelming consensus on the internet is that Chris King makes the best headsets. They work well and they last. Sounds good. Headsets don't cost all that much, relatively speaking, and, while a Chris King Nothreadset costs over $100, which is twice what a lower-end headset costs, it's still not the most expensive one out there. A pro-level component for not much over $100? Sold.
I thought getting a fork was going to be much simpler than what it turned out to be. That's because I didn't know as much about forks as I do now. Specifically: fork geometry. See, I thought I could just buy any old carbon fork and it would be good. So I went to Bike Island, which stocks components found on BikesDirect.com bikes, and looked at forks. Something I noticed was this thing called "rake". It differed between different forks. Hmmm. So I did some reading on the internet about forks, which lead to another term: "trail". Good thing I did. Because of my frame's head tube angle, the rake found on most forks would not give me the right amount of trail. That article said a trail of 57mm is ideal but that is what my last bike had and I could not ride hands-free for very long so I wanted a bit more trail. I used this handy trail calculator and found that I needed a fork with a rake of between 48 and 50mm to get the trail I wanted (time will tell if it makes much difference). Finding a fork with a trail like that was not much of a problem. For instance, both 3T and ENVE make forks like that. Finding a fork with a trail like that that wasn't over $400 proved to be more of a test. I finally found an online store that had a 3T Funda Pro fork with a 49mm rake (which made my trail 60mm) at a good price. Sold.
Fork (with crown race), headset, and frame. All I needed to do was put them together. It turns out that assembling all that requires specialized tools that is just not worth me purchasing. I dropped everything off at the LBS and the job was done the next day. If you get a headset installed on your frame, check out its orientation. If the headset's logo is aligned (e.g. the logo faces forward), your shop is paying attention to details (mine did). That's a good thing.