. . . the shape of shadows . . . morrows not yet made . . . drink from the cup of ice . . . drink from the cup of fire . . .
This ties in well with Rhaegar's "song of ice and fire", as well as the overall series title which Martin has indicated as containing several meanings.
. . . mother of dragons . . . child of three . . .
. . . mother of dragons . . . child of storm . . .
"Child of three" is very obscure. In a literal sense, one would have said that she was such a child due to being descended both from Aegon and Rhaenys and Aegon and Visenya. However, she claims her descent to be solely from Aegon and Rhaenys. Another possibility is that just as she is mother to her dragons, they will grow to be her protectors and she will be like a child to them. Yet another possibility is that Daenerys is not in fact Aerys's daughter, but Rhaegar's by an incestuous relationship with his own mother -- however, this is a fringe theory and not one we favor.
"Child of storm" is straightforward, given Daenerys Stormborn. On the other hand, it may refer to the turmoil and death her claim to the throne and her role in the war against the Others will bring.