Good introduction to Turing's work and complexity theory?
Some popular books are Computational Complexity by Papadimitriou and Introduction to the Theory of Computation by Sipser. If you are very ambitious, you can also try Computational Complexity: A Modern Approach by Arora and Barak, which is an advanced textbook.

I would recommend against the historical approach of reading old papers. While some old papers are a good read (some of Turing's papers, Shannon's 1948 paper), old papers tend not to follow modern conventions, have a somewhat different focus, are less streamlined than modern treatments, could contain weaker results, and (in some cases) wrong results or at least wrong proofs. A good textbook is much preferable.

Regarding Turing machines, while they have some theoretical advantages in some circumstances, they are poor models of actual computers. Indeed, even in the theoretical computer science community, algorithms are usually described in the so-called RAM machine model which conforms much more closely to actual computers. Even historically speaking, Turing's model wasn't the first universal computation model, though (if I remember correctly) it was the first one that Gödel found convincing (Gödel's own general recursive functions, for example, predates it).

On your way to complexity theory, don't forget to stop at recursion theory (sometimes called computability theory), which had been practiced for a few decades before modern complexity theory started in the early 1970s. The usual course sequence indeed starts at (very basic) recursion theory and progresses to complexity theory.