It is imperative that climbing ropes are in excellent shape so they don't split while someone is climbing.
How do I properly inspect the rope when I'm about to start a climb?
I would inspect your rope long before you get to the crag, rather than just before you're about to use/need it.
Rope inspection is essentially going over the whole length of the rope, looking for grit or evidence of wear in the outer layer. One thing to note, the inside of the rope might have been strained and this isn't easily visible from the outside - sometimes you can feel the outer layer sliding over these areas.
Rope has a shelf-life - but this also depends on how often you use it. Another rule of thumb to follow is that if you have taking one major outdoor lead fall on the rope (from above protection), then you retire it. You can take several minor falls before retiring it. Top-rope falls should almost have negligible effect - unless there was a lot of slack in the system for some reason and the climber fell more than a couple of metres or so.
If, at any time, you are not confident in your rope - replace it. The price is worth a lot less than a climbing injury.
It is good while checking the rope to squeeze it between the fingers to try to feel any voids inside the sheath. The absence of a void does not mean the rope is safe but the presence of a void justifies retirement.
Ropes typically have a climbing life of 3 years.
Another point of inspection is to be certain the rope is long enough for the climb. Tie a knot at the end of your rope to prevent the belayer from paying out all the rope causing the climber to fall to the ground without a belayer to arrest the fall.
A few points that haven't been added here yet:
As a side note, I'd also suggest reading "Accidents in North American Mountaineering" as a great way to keep on top of what actually goes wrong in climbing. It's illuminating to see what mistakes really are causing deaths in climbing. The book summarizes and analyzes climbing accidents each year.
Ideally, Inspect it indoors, before you leave for Outdoors. :) Look out for tangible signs like wear n tear on outer layer of rope, cuts, brazes etc. For the inner layer, it should be intact as with outer layer. If its moving on its own, should ring bells to your head!