The correct behaviour is NOT to trust the computer.
When I log in at one, if I can't insert a USB stick with my own copy of Firefox on it for browsing, I'll load theirs up, but will make sure it's updated first to the latest version, for security (or whatever other browser they might use).
I'll then check the running tasks on the machine and see if anything looks suspect. This is harder for someone non-technical to do as you may not know which processes are part of Windows etc, but it's a step.
For your actual password, if you're worried about keylogging, you could always just type a letter, then in an open notepad type a bunch of garbage, then the next letter, and repeat. Unless their keylogger is sophisticated enough to be application specific, of course.
At that point, you'll want to consider two factor authentication. Get either an SMS or in-app message with a code that you type in (Gmail and more can be set up for this), or a QR code that your phone scans on the screen (Whatsapp web does this).
If you're getting really fancy you could stick an operating system on a USB stick, prebuilt with the browser of your choice etc and then boot the machine to that, but it depends on you being able to get into the BIOS, or what other admin restrictions they've placed on the computer (or if you can even reach the USB port).
Afterwards, clear the cache, cookies etc of the browser, and I tend to reboot the computer when I leave as well, as some internet cafes are set to reinstall everything from scratch on reboot, wiping any trace of me having been there (I once worked in an internet cafe where we did this).