The Encyclopedia of Ancient Christianity on this topic points us to The Shepherd of Hermas. This work is dated to about the same time as the Didache (last first or early second century) and bears some similarities with it, and describes more specifically how a Christian is to fast:
First of all, be on your guard against every evil word, and every evil desire, and purify your heart from all the vanities of this world. If you guard against these things, your fasting will be perfect. And you will do also as follows. Having fulfilled what is written, in the day on which you fast you will taste nothing but bread and water; and having reckoned up the price of the dishes of that day which you intended to have eaten, you will give it to a widow, or an orphan, or to some person in want. (Similtude 5, ch. 3)
We quickly notice the limitation to only bread and water, but the fast is not merely self-denial with respect to food, but an emphasis on purifying oneself spiritually and showing generosity to the needy.