I actually played Mods more when I was newer to Civ than I do now. I really enjoy the base game and setting challenges for myself. One being to try to get as good at the game as possible.
Quite a few features of CFC here for the base game that you may not be aware of or just have not taken advantage of, or unsure of the merits.
Here's a few things to consider:
1) Strategy & Tips forum - learn to play the game better and challenge yourself. Lot's of forum games are posted including Noble's Club, Immortal U, etc. Post questions and join in discussions. Run your own forum game to learn or challenge others.
2) Game of the Month - Monthly games created by the staff for BTS, Warlords and Vanilla. Compete for fun against other and try to win awards and ranking. ( I have some links in my sig, but I'll post it here as well)
GOTM
3) Hall of Fame - Basically, a CFC database to submit games and see them ranked for all time. A permanent and more dynamic extension of the HOF on your computer. Monthly Gauntlets and Challenger Series are offered to provide some friendly and personal competition. Try to become #1 in a particular category and also achieve QuattroMatster status. A new Challenger Series just start up, and I posted an announcement right in this very forum recently, as seen here:
Announcement thread
Link to HOF forum in my Sig below
4) Succession Game of the Month - A sub-category and sub-forum of the GOTM, the SGOTM is really more of a quarterly series of games played by teams. SGOTMs are really fun as you play in a team environment on maps/scenarios specifically designed to create challenge and fun for the teams. Skill level doesn't matter, as there are usually always a mix in each team. Great way to learn. Top 3 teams earn Laurels.
SGOTM
There's also the Stories & Tales forum and Succession game sub-forum. Check them out.