I work in (server) operations and it is not very uncommon for me to parse logs and try to make sense of some kind of metric. For some odd reason I have grown quite fond of it.
Assume that I have almost no mathematical skills beyond arithmetics. What would be the most natural order of study for me to gain a basic understanding of statistics; enough to help me in my work?
If you pick up a book on Engineering Statistics, you might encounter some basic concepts that don't require any advanced math at all.
When processing logs statistically, you're typically basically looking at one thing: the occurrence rate of some event (say, a process crash).
There are probability models that treat discrete events as point processes (for instance, Poisson processes). And consequently, there are statistics based around these models.
For instance, if an event such as a process crash is a truly random event, then it should follow a Poisson distribution. Then all you have to worry about is occurrence rate. But if you find that your inter-event times are not Poisson, say, that they are uniform, then this means something bad is happening with regularity! Hence, a simple probability model determines how you investigate the problem.
Engineering statistics covers this easily, and you can probably find many resources online.