There is a book by Rosen called "Number Theory in Function Fields". I am reading it now, and it is a really nice book- it is readable and the analytic parts of the theory are very well explained to my opinion.
However, it lacks the algebraic background you are looking for. For algebraic background and more detailed proofs of algebraic facts, I think a great book is "Algebraic Function Fields and Codes" by Stichtenoth. The first chapter of it covers up most of the algebraic background needed, without any special prerequisites (you may want to have some fimiliarity with definitions from ring theory but not more than that is actually essential), and it is very well motivated. And you can read further if some parts of Rosen's book are not clear enough. Note, however, that this book contains some chapters about the application of the math of function fields to coding theory, which probably would not interest you, but you can skip them without losing anything important.
Edit: Looking more carefully into the chapters you mention, maybe the 4 chapters of Rosen should be enough - they seem to concern only number theory in $k=F_q[t] $ which (Its field of fractions) is the simplest function field. The next chapters in Rosen concern extensions of $k$, and so maybe won't be that related. Same goes to Stichtenoth. Maybe you should try to read it in parallel with the book you read now, and see what you actually need to learn.