Christine, your question is a classic illustration of the age-old tension between the sovereignty of God and the free will of man! The problem is that we seem to find both in Scripture.
I begin by quoting from the notes of the NIV Study Bible (on Ex 4:21,p. 92).
Nine times in Exodus the hardening of the pharaoh's heart is ascribed to God (here: 7:3; 9:12; 10:1,20,27; 11:10; 14:4,8).
Another nine times the pharaoh is said to have hardened his own heart (7:13-14, 22; 8:15,19,32; 9:7,34-35).
The pharaoh alone was the agent of the hardening in each of the first five plagues. Not until the sixth plaque did God confirm the pharaoh's willful action (9:12) as he had told Moses he would do (Ex 9:16).
Thus it would appear that the pharaoh initiated his response to Moses' request with a challenge to God, illustrated by his strong word of defiance in 5:2, "Who is the LORD [in Hebrew, "Yahweh," God's personal name], that I should obey him and let Israel go? I do not know the LORD, and I will not let Israel go."
Here is a basic theme in Exodus. As a result of the plagues and Moses' long suffering relationship of petitioning the freedom of his people, the pharaoh came to know who God was - as did all the people of Egypt and even of the surrounding countries.
But getting back to your fundamental question, perhaps the best biblical commentary on the issue is found in Romans 9. As Paul works through his anxiety over Israel's failure to respond to Jesus as the promised Messiah he describes their privileges as God's chosen people.
Then, he notes that not race but faith is determinative in being Abraham's true offspring. Perhaps as a surprise (v. 11) he says "in order that God's purpose in election might stand," we are not saved by works - by what we do - but by God's grace and mercy.
To the question, "Is God unjust?" he replies that God can have mercy on those whom he will. At this point he brings up the illustration of Pharaoh, and concludes categorically, "God has mercy on whom he wants to have mercy, and he hardens whom he wants to harden."
To those who object to what seems an arbitrary action, he responds, "Who are you, O man, to talk back to God?" End of discussion. Or is it?
He continues on to show how Israel did not pursue righteousness by faith but by works (v. 32). He goes on to show God's mercy to the Gentiles and how one can receive his grace and be saved by confessing with one's mouth that Jesus is Lord and believing in one's heart that God raised him from the dead (10:9). Here is the Gospel, calling on people to respond!
He concludes (11:32), "God has bound all men over to disobedience so that he may have mercy on them all." And his final word is a paean of praise to God for his wisdom, righteousness, and mercy, to whom we should give glory (11:33 following).
Another possible commentary on the experience of Pharaoh could be Romans 1:18 following. Here we see people who suppress the truth by their wickedness, thus being without excuse. Claiming to be wise, they become fools (v. 22).
Then twice, Paul says that "God gave them up"; he let them go their own way without restraint. Following this pattern, we could say that Pharaoh had five plagues to see God's glory; when he didn't respond, God let him go on his way.
There is a climax in Exodus (9:16). God called Moses to confront the pharaoh. He threatens to send "the full force of my plagues against you...so you may know that there is no one like me in all the earth." He mentions that he could have wiped out the whole nation of Egypt.
Instead (as an act of mercy, really) he states, "I raised you up for this very purpose that I might display my power in you and that my name might be proclaimed in all the earth." Everything God does is for his own glory. For a sinner to say this, it would be an illustration of pride. But God is no sinner. The greatest blessing he can bring to us is for us to come to appreciate just who he is. This was the focus of Jesus' life (Jn 17:3). And Paul says that all we do, "do it all for the glory of God" (1 Co 10:31).
When it comes right down to it, Christine, there is always mystery when we get down to God's wisdom, nature, and ways. The above is the best I can do for you at the moment. I trust you will find it somewhat helpful. I would suggest that you do your own study of Romans 9-11 and ask the Lord to help you understand more fully this complex and profound section of Scripture.
Blessings on you and your small group.