F-stops deal with doubling/halving the amount of light hitting the sensor. Everything revolves around twos.
With the shutter speed, it's easy to understand, as you say. Every shutter f-stop is (roughly) half/double the amount of time as the previous one. Personally, I don't even bother paying attention to the numerator ("1/") part of the shutter speed; I've drilled it into my head that bigger denominator = faster = less light = darker exposure.
Note that shutter speeds aren't exactly doubles/halves. I think that this is just because manufacturers think people like to see "round" numbers. At the fast end, that means 1000, 500, 250. At the slow end, you need more accuracy, so you have true halving of speed (1, 2, 4, 8). Then, they have to make the numbers meet in the middle, so they start to fudge the numbers a bit (15 is almost 8 * 2, 125 is almost 60 * 2). (I'm a programmer, so personally, I'm fine with seeing a shutter speed of 1/1024s :-) )
Aperture is a bit trickier. Double the light means doubling the area of the aperture, which is where the squares/roots come into play (Area of a circle = pi * r^2). That's a pain to mentally calculate, but there is an easier trick to consider: every two stops represents a doubling (or halving) of the aperture's f-number:
1, 2, 4, 8, 16, 32, 64.
If you know those, then you can guesstimate the in-between stops by calculating slightly less than half of the average of the surrounding f-stops:
1.5 -> 1.4, 3 -> 2.8, 6 -> 5.6, 12 -> 11, 24 -> 22, 48 -> 45.
As with shutter speed, bigger number = smaller aperture = less light = darker exposure.
Something similar happens with ISO. Each doubling of the ISO value represents a stop, which you can trade off (with consequences) with stops of shutter and aperture. Note that this transition is reversed though: bigger number = more sensitive = more light = brighter exposure. The common ISOs are:
50, 100, 200, 400, 800, 1600, 3200, 6400, 12800
And just to be complete, there's another similar scale with flash power:
1 (Full power), 1/2 power, 1/4 power, 1/8, 1/16, 1/32, 1/64, 1/128
This is very much like shutter: bigger denominators (forget the numerators) = less power = less light = darker exposure. (Note that true powers of two is fine here).
Honestly though, I don't bother with any of these mnemonics myself. I usually do "three clicks of my control wheels on my camera" when I want to go up/down one stop. (My camera, and many others, set one click of the control wheel to be 1/3 of a stop.) The absolute numbers aren't usually as important as the amount of change relative to "where you are now".