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.