I'm tinkering with a Server 2008 R2 server that has a 16 GB C: drive. After running Windows update, I only have 340 MB of free space left. C:\Windows\Assembly is taking up 8 GB and C:\Windows\Installer is takiing up 2 GB.
What is the best way to make more room on my C: drive?
Honestly? 16 GB for C: on Server 2008 R2 is woefully insufficient, as you've already noticed. Extend it with something or, since you're just tinkering with this server, buy a bigger disk and put it in. MS recommends at least 40 GB.
(ah, missed the part about only allocating 16GB to the C: partition to begin with. that'll cause problems. my response is for more general space issues not related to what amounts to a misconfiguration.)
The first thing to do is to step back and decide whether or not this makes sense. Do you likely have a full drive's worth of actual data + applications or is something, somewhere going unchecked and causing issues for you.
If you feel that it does makes sense, then there's not a whole lot you can do besides move some of that data off of the drive. Sure, you can compress some folders and their contents and that might gain you a bit, but it's not going to solve the problem in the long run.
If you don't think that you should have a full hard drive at this point, you'll need to do some more research. Among other possibilities, there might be some extremely verbose logging that is causing the space issues.