Yes, that's nearly it. The key thing is that, with respect to ground, the AC signal moves between the rails (well, the rails plus a diode drop). So yes, when the AC input is at its most positive, the first capacitor is indeed charged via the first diode. When the AC input is at its most negative, the negative end of the first capacitor is taken negative, and the second/bottom diode is able to conduct, and this charges the smoothing capacitor (the second one).
This circuit is a classic arrangement - I didn't invent it, sadly. It's fine for the small currents we're using here...
As an aside, one point that isn't always appreciated at first is that the smoothing capacitors supply energy to the load for most of the time. The current from the transformer (via the rectifier diodes) only flows for a brief time. Therefore, the peak current that flows during this time is actually much greater than the average DC current that is supplied to the load - perhaps 4 or 5 times! That's why the wiring between transformer, rectifier and smoothing capacitor must be short and heavy. It's also why you should take the output from the smoothing capacitor, not the rectifier...
All the best,
Mark