Having recently finished my math undergrad I audited a course based on Davenport and had a reading course using Apostol, which to use depends on what skills and smaller results you want them to come away with. The things you mentioned like big-O and summation are given a pretty thorough treatment in Apostol. I certainly wasn't cheated and really appreciated having some practice with the skills. It also had enough material for you to have some flexibility.
But, it might seem like too much of an undergraduate text (it introduces the definition of a group before it talks about characters). It also gives a pretty elementary proof for Dirichlet's theorem. Which you may not want.
A book not mentioned that also has a lot of topics and is nice to learn from is Additive Number Theory by Melvyn Nathanson. The material here is very different from that of the other two, but still worthwhile and accessible.
If I had to pick one, I'd go with Apostol. It was so readable and I felt like I got a great foundation in the ideas and skills of number theory.
