Through the progress of the play Macbeth, do the three weird sisters gain power?
In Shakespeare's Macbeth, the three weird sisters do gain power, or more specifically, are given more power by Macbeth, or are allowed more power by Macbeth, as the play continues.
The witches' manipulation and equivocation is at first met with hopeful skepticism by Macbeth. He wants to believe but is skeptical. But once he becomes king, his ambition deepens and he moves from skepticism to wishful thinking. In short, he is willing to believe almost anything that suggests he will remain king and create a dynasty in Scotland.
For instance, the prediction that he cannot be harmed by anyone born of woman amounts to invincibility. Macbeth embraces the prediction, even though it is so unbelievable. At times Macbeth rationally seems to know this prediction is too good to be true, but soon after he goes back to believing he is invincible.
Macbeth even seeks out the witches and goes to see them in Act 4.1. The witches, in fact, serve as his advisors--advisors to the king. Macbeth certainly interacts with the witches more than he does his wife in the latter stages of the play.
The witches serve as the catalyst for the plot of the play, and their power only increases as the play progresses.