As a frequent XOTM player, I'll take this one.
First of all welcome to CFC
There are 2 different games played each month. The first is the Conquest of the Month (COTM) and runs from the 1st to the last day of the month. You'll need to have the most current release of C3C (1.22) loaded to be able to play. Just download the game and start playing. There are a couple of rules (like no reloading/replaying turns) that you'll have to follow, but in general it's a good way to learn how to get better. The most recent one became available about 2 weeks ago and is COTM 11.
The second game is the GOTM, also known at the Classic Game. This can be played on either PTW or Vanilla at the final patch levels. The PTW version is just download and play. The Vanilla version requires a downloaded mod pack to include new units and tribes that were added with PTW. This one runs from the 15th to the 15th and GOTM 42 started this morning.
For both games, a spoiler thread will open about five days after the game begins and will normally cover the ancient age. People will discuss their strategies and tactics and you'll be able to read what others did and post what you did and ask why/how they did what they did.
Sometime before the end of the 30 period from when the game started you'll want to submit your game (there's a link on the download page) so it can be included in the results. Don't worry about how you do, lots of losses get submitted.
There is also a Quick Start Challenge (QSC) where you keep a detailed log of your first 80 turns and submit that along with your 1000BC save. You'll be able to compare your start with the other players and learn new techniques to improve your opening plays.
The are 3 levels of play. For a newcomer, you'll probably want the Conquest level. That ususally includes either player bonus or AI handicap (or both) to help you with the early game score. The other two levels, Open (no pluses or minuses for anyone) and Predator (which handicaps the human and helps the AI) are also available.
EDIT: X-Posted with Ginger Ale