You're in a difficult situation: you're interested in material that is usually taught at the graduate level, but you want a book written at a lower level. That's a common problem for students in mathematics, and there's usually no easy fix. Most undergraduate-level books on computability only go through the halting problem, and then stop. Cutland's textbook, which I think is a very good undergraduate book, is an example of this.
Soare's book is the standard textbook in computability at the moment (if there is such a thing as a standard textbook). It's a beginning graduate-level book, which means it doesn't require any previous knowledge of computability, just a significant investment of effort to learn the material. Most graduate-level books will require a lot of careful reading and rereading, along with working exercises, to learn the material. That's simply the way things are.
Cooper's book is written in a way that makes the material feel much more accessible, like an advanced undergraduate level. I think it does a good job of presenting the way that computability theorists think about the area, and the book has a much larger selection of topics than other books written at a similar level. However, it does this by omitting many details, which you would have to work out for yourself. There is a risk, with books of this sort, that you may have a false confidence in your understanding of the material. Soare's book is much more fastidious about details. In the end, using Cooper's book would require just as much effort as Soare's book, except that you would have to work out the proof details for yourself.
My recommendation would be to read Cooper's book to get an idea what's going on, and to get a general idea, then dig into Soare's book to learn the proofs in detail.
There is also a classic text by Rogers, which is well regarded. Although the book is a little old, the material on computability and priority arguments is very well written and still perfectly applicable today.