The Old Testament is foundational; the New Testament builds on that foundation with further revelation (details) from God/Jesus
Then why do they seem to be books about Gods with completely different characters and approaches to solving problems?
The Old Testament establishes principles that are seen to be illustrative of New Testament truths.
Then why does the New Testament immediately abolish just about everything the Old Testament commanded?
The Old Testament contains many prophecies that are fulfilled in the New.
The New Testament is a series of books. The gospel writers would have known all about the prophecies.
The Old Testament provides the history of a people;
the New Testament focus is on a Person/individual.
In what way does this explain the contrast in their views of God?
The Old Testament shows the wrath of God against sin (with glimpses of His grace); the New Testament shows the grace of God toward sinners (with glimpses of His wrath).
In the Old Testament, God's answer to every problem is to curse and kill. In the New Testament, God's answer to every problem is to love and forgive. For example, in the New Testament, God is confronted by a problem. His creation is growing wicked. The world is getting out of control. His solution? To incarnate himself as a human, preaching a message of peace and love, meekness and submission.
Is this the message God gave us in the Old Testament when confronted with a similar problem? Well, no. His reaction then was to drown almost every living thing on earth.
Something of a difference? I have a little trouble seeing how the Old Testament, full of genocide, slaughter, rape, murder and brutality, all sanctioned and indeed often ordered by God, is "what humans were ready for at the time" and that His real message was to do good to other people, bless those that curse you, and forgive when provoked, again and again.