My objective is to buy an Arduino Microcontroller, start with the basics and keep tearing-down and rebuilding until I make a Quadcopter.
Is the Arduino Mega ADK rev3 a good choice?
I want to be able to track my unit over a fairly large distance eventually, so I would like to choose a micro controller that already has shields on the market for either GSM or Mesh Networking.
My quadcopter will be fairly powerful and will carry a GoPro. I do not mind investing extra cash in more powerful Axi motors and larger props to handle heavier weight.
Which Arduino products (i.e. shields) should I buy at the same time?
A lot of people have put a lot of effort into solving this problem so unless you have a particular desire to reinvent everything yourself I suggest you start by researching diy drones and ardupilot. I understand there are some Arduino-based clones which incorporate the various sensors needed to control a drone, and there are also open source software projects to manage the sensors, implement the analysis and control feedback loops and comms systems and so on to make this sort of thing work. They are very sophisticated these days and creating your own equivalent solution from scratch would be a huge undertaking.