What is the significance of the statement "With this you will know that I am Hashem"?
Seforno: Pharaoh would now learn that Hashem (via the Name Havayah) manipulates the laws of nature, to destroy the Nile which is naturally indestructible. 1
Rashbam: It is the answer to Pharaoh's statement (in Shemos, 5:2) "I do not know Hashem" - Now he was about to receive practical demonstration as to who Hashem was.
Riva: Pharaoh [made himself a god, and] said "the river is mine, and I made it." Now you will see that I am G-d, and not you, for I will strike the river, and you cannot save it.
Why did Hashem begin the plagues with the turning of the water into blood?
Rashi: Because, due to the fact that there is no rain in Egypt and they rely totally on the Nile overflowing its banks for their water supply, the Egyptians worship the Nile - Hashem struck their god before striking them.
Why did Hashem tell Moshe to say "with the staff in my hand"?
Hadar Zekenim: Pharaoh feared the staff.