The battery voltage measurement is done with a simple resistor divider and one of the ADC pins of the PICAXE. With a 3:1 divider you can measure 0-20v since the PICAXE pin only sees 0-5v over that range.
You can measure the battery voltage by looking at any of the nominal "+12v" lines, the only caveat is that if you want the accurate
battery voltage you have to run leads directly
to the battery terminals, otherwise the voltage drop in the bike wiring factors into the equation. However, all you really need to know is if the battery voltage is less than about 12v (indicating that the alternator is faulty) or over about 15v (indicating the regulator is faulty) or in the normal operating range (around 14v with the bike at 2000 rpm or more). If you know the voltage drop on the leads you are connected to (and that drop doesn't change much), you can compensate for it in software.
You can then light one or more LEDs to show battery condition (depending on how many pins you have), or you can use one LED and flash it for high and low voltage conditions. Or you can use two LEDS connected to one output pin with opposite polarities, so that a high output gives you green (normal) and a low output gives you red (over or under voltage).
As for how you learn this stuff I guess it depend on if you've ever had any experience with electronics. What's 2nd nature to me by now might be a complete mystery to someone who has never used a soldering iron. Basically you learn by doing. There are "starter kits" available for most microcontroller systems. I think the PICAXE system is probably one of the easiest to learn, and it's used in schools over in the UK. Their website http://www.picaxe.com/
has all the info you could ever need to get started and they have "starter kits" for simple projects.