If you already have a desktop or a laptop, great. When developing for the Arduino you will need a fully functional desktop operating system so that you can use the Arduino IDE to write, compile and upload code to the Arduino. If you don't have a computer already you can use and don't want to spend a lot of money, the Raspberry PI 3 is a great option currently available. This document will point you in the right direction for all the hardware and software you need to get started. The best part is that all the software is free and the hardware is cheap. If you already have a monitor with HDMI input, you don't even need to buy another monitor. If you don't, any TV with HDMI will work as long as it isn't 4K. These days, there is very little difference between TVs and computer monitors except the price.
This PDF is the language reference sheet for the Arduino. We'll be starting out by going through the core language features of the Arduino before getting into any necessary libraries.
The focus of RetroLCD.com is teaching programming through game development. This Video Game Plan Form is a recreation of the one used by the people who created Asteroids. Atari Asteroids: Creating a Vector Arcade Classic Use this document to start thinking about what kind of game you would like to make. The limited space on this document is deliberate. Remember, we're developing for an 8-bit computer with very limited memory. Start with a simple idea. You can always expand on it later.
A quick run through the fundamental programming constructs we'll be using in the game programming curriculum. This goes along with the Arduino Reference Guide. Note that it does not cover the Arduino specific things like reading and writing digital pins. We're just getting started. We'll cover those later after we learn why we would need them for game programming.