First I highly recommend a copy of the St. Augustine's Prayer Book. It covers this topic and many more really well. It's an excellent guidebook for high church practice in everyday life.
The beauty of Anglicanism is, how to perform that act of discipline and self-denial is not set by the church. You are free to determine what that might look like for you.
The traditional form is to fast from food, which means reduce your consumption (one full meal, one light meal, and skip one.) Or abstaining, meaning no meat, but fish is okay. Traditionally you give that money you saved on food to the poor. Unless it's Ash Wednesday or Good Friday, where a "hard" fast (eating nothing) is more appropriate.
Or maybe you fast from something else, like Netflix or whatever, and give that time to God instead. I would encourage trying the traditional approach at least for a season, it is Advent after all!
Given your background and devotional tendencies, it sounds like you may be suited to what's called Anglocatholic leanings. There may be churches in your area that lean that way, who use incense and have corporate rosaries, for instance. I would suggest this little manual, that is a very catholic style guide to prayer for Episcopalians:
Also, this parish in Philly is a great example of just how Catholic the Episcopal Church CAN be. Most parishes won't be quite this Catholic tending, but I post just to show that you wouldn't have to leave Catholic practices with you.
Here it is on Amazon:
St. Augustine's Prayer Book - Amazon
If you'd rather not buy from Amazon I believe it's available through the Forward Movement website aswell.