My habit with server logs is: review them regularly, and investigate/resolve the issues I find. I do this proactively - not waiting until the users are howling about a system outage. The main reason this is effective, really boils down to a couple of old sayings:
A stitch in time saves nine.
Obviously if you're solving issues while they're small, you're ahead of the curve, and users/management will have less reasons to yell at you; that's a good thing.
Practice makes perfect.
I think this is the greater advantage to the sysadmin. By getting in there regularly and proactively reading the logs, you're gaining experience and familiarity. You're learning what those cryptic log messages mean - and which are trivial, and which are a big deal. The process of investigating messages you don't immediately understand (which will be a lot of them at first!) teaches you a lot about the internals of the OS and the apps running on it.
Usually when I get a new system to manage, it will have quite a few errors in the log, many which recur fairly regularly. The prior admin often shrugs them off with something to the effect of "not real sure what that's about, but the users never complained, so I didn't consider it broken enough to fix!"
My goal with such systems is to revisit the logs weekly until I have solved or understood every new error that comes up; then relax my log reviews to monthly. Clean logs are easier to read!