Controlling a sliding tray - beginners project
Am I barking up the wrong tree and shouldn't be attempting to find a solution for this with an Arduino? 


Not at all. For example see Reprap - a 3D printer using an Atmega1280 as the controller. As it says here: 


  Arduino is an open source project that has created an easy and powerful microcontroller board based on the Atmel ATmega168. It is the brain of the RepRap electronics.


If you can make a 3D printer with an Arduino, then you can make your sliding platform gadget, which sounds simpler if anything.




  Are there existing projects that basically do what I'm hoping to achieve?


Reprap, for one.




  Are there books that are considered the "bible" of Arduino development?


This is a broad question, and your original question already has a couple of "flags" that it is too broad. I suggest you narrow it down a bit.

There are a number of books about learning Arduino. My advice is to start doing some simple projects (simpler than the platform initially) to get the hang of compiling, uploading, and generally working with microcontrollers. 



Also try Google and YouTube. There are a lot of Arduino projects out there.