If I understand it well now, what you want are books about basic number theory with a good algebraic foundation. I can recommend the following:
Elementary methods in number theory (2000), by Nathanson. Graduate Texts in Mathematics. It starts low, but it reaches quite high.
Elementary Number Theory with Applications (2007) by Koshy. Elsevier. Truly basic. Not very very algebraic, but a really nice textbook.
Algebra and number theory by Andrew Baker. Online notes. Fairly basic.
A computational introduction to number theory and algebra (2005) by Shoup. Cambridge University Press / Online Free Version. Despite the title, I think it satisfies the conditions you are looking for.