Some process is slowly taking over disk space...how do I find out what this is and how do I deal with it?
It’s a combination of the page file (virtual memory) and the hibernation file that gets written when you put your computer to sleep. Of course, other processes may also leave temp files that get deleted when you restart. There isn’t really much you can do to improve that without affecting your computer’s performance. 

You can get a utility like WhatSize, Daisy Disk or similar to visually inspect the contents of your hard drive and see where is the “abnormal” file or directory. 

You can also use Monolingual to reduce the size of your executables and remove unwanted languages and files.

Update: 

The virtual memory file is located in 

/private/var/vm


and the hibernation/sleeping file in 

/var/vm/sleepimage


Please not that you can’t (and shouldn’t) delete the vm file while OS X is running.