I use Uber in London and am going to visit Amsterdam in the summer. I know Uber serve in Amsterdam, but do I have to register anything in particular in my Uber phone app or website to be able to hail and pay for a cab in Amsterdam (or any other city)?
No. I am also registered with Uber in London and have successfully used it in Amsterdam, Cape Town, and other cities. It's the same account, same process.
Of course, you'll be billed in local currency - and don't forget you may have to pay roaming data fees on your phone contract.
Being an app-based service I would assume that all you need to use Uber worldwide is the app and an internet connection on your phone, be it WiFi or mobile internet. This question on Quora indeed says it should work seamlessly using your everyday Uber account.
A small piece of advice would be ensuring you have mobile internet on the phone with the Uber app, so to be receive the notification in real time, without having to rely on WiFi hotspots. Indeed this FlyerTalk thread mentions that having mobile internet at all times allows you to find your driver on the map in real time, thus avoiding unpleasant issues.