I have taken two introductory courses on logic. One was an undergraduate level and the second one was at the graduate level. Both used a set of notes written by the instructor. I'm thinking about graduate school in the field and would like to spend my next summer reading more about logic.

What I'm looking at is a good list of rigorous graduate level textbooks on the most important topics in logic that would get one started and which would provide a good foundation for graduate level studies. I have the whole summer and my senior year to prepare.

Which books would you recommend? My plan at this point would be to read a rigorous graduate level textbook on basic logic and then one on model theory. The ones that I've looked at are:

Manin, A Course in Mathematical Logic for Mathematicians

Marker, Model Theory: An Introduction

Would this be a nice place to start or are there better options?