How does Prospero change from the vengeance at the beginning of The Tempest to forgiveness? Include direct quotes for a better understanding.
It seems that you are assuming that Prospero is acting in service to revenge or vengeance at the opening of the play, which isn't necessarily true. If vengeance is his motive, why doesn't he just sink the ship and drown the passengers and be done with it? Instead, once Propsero confirms from Ariel that the tempest has wrecked the ship as he intended, he asks:
But are they, Ariel, safe?
Not a hair perished...
...Safely in harbour
Is the King's ship...
Ariel, thy charge
Exactly is performed.
So, Shakespeare goes to some trouble here to let the audience know, from the very beginning, that even though he has cause to retaliate and revenge, he is concerned for the safety of those on board the ship.
Prospero does not, as many of Shakespeare's characters do, fill the audience in on his plans and schemes by way of soliloquies, but he does seem to be acting all along by a premeditated scheme. He says, when Ferdinand and Miranda fall in love at first sight in Act I, scene ii:
It goes on, I see,
As my soul prompts it.
Which indicates his power over the events of the play. There is no moment that Prospero does not have ultimate control over, so, when he gives his forgiveness and requests to be re-instated to his Dukedom at the end, we can only assume that this forgiveness and reconciliation was his plan all along.
He does put Alonso, Antonio, Sebastian and Gonzalo through some hardship before the final reconciliation at the end, but, since Gonzalo has only ever been his friend, it is hard to see the motive for this as vengeance. He stops the killing of Alonso, and once his major scheme, the uniting of Ferdinand and Miranda, is complete, he quickly reunites all parties for the final moments of the play.
As I read the play, this forgiveness showered upon Prospero's brother and Alonso is not some change of heart that happens through the course of the play, but has been Prospero's plan from the opening scene of The Tempest.
Does Prospero have a plan? Yes he does.
I believe that there are three reasons he causes the the storm and brings the ship safely to his island. Despite all his power, he is powerless to get off the island and needs the ship for that purpose. Miranda is a young girl and Prospero wants to find a husband for his daughter. And, yes, he desires vengeance on his brother and those who helped him.
When he says that everything is going well, he is speaking specifically of finding a husband for Miranda. As he had hoped, the young people are attracted to each other. That Ferdinand is Alanso's son is ironic and this is not lost on Propero.
What vengeance Propero has planned for Antonio, Alonzo and Sebestian we will never know. Was he planning to avenge what these men did to him and Miranda? Definitely.
In Act V, scene 1, Ariel reports back to Propero concerning the plight of these men. Ariel tells him that these men are really suffering and even he feels pity for them. When Propero questions this, Ariel tells him if he were human his affections would become tender. It is at this point that Propero changes his mind.
Though with their high wrongs I am struck to the quick,
Yet with my nobler reason 'gainst my fury ,
Do I take part. The rare action is
In virtue than in vengeance.
It would seem that Propero did indeed have vengeance in mind but because Ariel, a non human, can pity the plight of these men, then they must really be suffering. From what Ariel has told him, they are all suffering including Gonzalo, a good man who helped Propero when he and his baby daughter were set adrift.
Propero has great power but he learns a great lesson. It takes a big man to choose not to use that power in a negative way. Vengeance is easy, forgiveness is much more difficult.