I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So far I've seen Lawvere Theories show up a lot.] What's the best way to learn them both?
[EDIT 2: What books (if any) introduce them both in detail?]
Here's my motivation: I'm fascinated by how different areas of Mathematics (and systems of Logic), particularly in Abstract Algebra, relate to one another and by how crucial certain axioms are in those relationships. I know this is quite vague but I've been chasing this stuff around for about a year; I know what it is I'm after when I see it, but I'm not knowledgeable enough (yet) to pin it down precisely. Category Theory & Model Theory (as well as Non-classical Logic) seem to hit the spot frequently.
EDIT: To make things easier/more precise, think of a kind of "mathematical KerPlunk," where the sticks are axioms and the marbles make up a (system of logic or) mathematical structure (with, say, red marbles for theorems, blue for definitions, etc.). If you remove (or change) certain sticks, what falls and why? Which marbles move? Do any change colour? Compare what you get with what you started with. How does the 'new' structure fit into the bigger picture? What are its 'neighbouring structures'? That's the kind of thing I'm interested in.
EDIT 3: Reverse Mathematics looks highly relevant but it's new to me. Is there any way I could get there via CT & MT?
EDIT 4: Suggestions on how to improve this question are welcome. I think Universal Algebra is relevant but I've replaced its tag with the Topos Theory one to narrow things down.
Best Answer
I've actually taught myself a fair bit of category theory and am currently learning model theory. My own historical route is probably not what I'd recommend: Robert Goldblatt's "Topoi" and Adamek et al. "Abstract and Concrete Categories", followed by Lawvere's "Sets for Mathematics" and Steve Awodey's "Category Theory". Probably doing that list entirely in reverse would be a gentler and saner approach.
I found the books on topos theory helped me a great deal, because I find set theory reasonably natural (ha) and it helped me relate categorical ideas to a category I had a fair sense for. I also made a project of doing all of Awodey's exercises, which are accessible with a relatively minimal grasp of set theory and abstract algebra. Also, whenever I ran across an interesting theorem that I could understand the meaning of, I stopped and tried the proof myself.
For model theory I'm using a combination of Goldblatt's book above, Chang & Keisler's "Model Theory" 3rd ed. and David Marker's "Model Theory: an Introduction." Model theory doesn't suit me as well as category theory, though, so I can't really recommend how to approach the material. I can say that Marker's a bit friendlier than Chang & Keisler. Multiple sources and doing lots of exercises doesn't hurt.
Hope this gives you some leads.