Are you familiar with a general treatment of first order logic, including the completeness theorem? If not, I suggest starting there, for which I'd recommend:

- Chiswell & Hodges, "Mathematical Logic" - This is very clear and careful and gives lots of motivations for introduced concepts, but moves somewhat slowly (introducing FOL in three stages, first propositional, then quantifier-free and then full FOL)

- Leary & Kristiansen, "A Friendly Introduction to Mathematical Logic" (available online for free) - this moves somewhat faster and skips "lower" logics such as propositional logic and uses a different proof system. If you read up to Löwenheim-Skolem, you already have a little bit of Model Theory (the rest of the book is more about computability and logic).

If you've already done FOL, I would recommend:

- Kirby, "An Invitation to Model Theory". This doesn't presuppose much more than FOL (and even recapitulates it briefly) and some basic familiarity with undergraduate math concepts (e.g. groups, fields, vector spaces) and explains everything very carefully. I also think it has great exercises.

Introductory model theory has no particular prerequisites beyond elementary set theory and logic, but it does demand a fair amount of mathematical maturity. If this is your first exposure to higher math, you're probably better off starting with another topic. Real analysis or group theory would be the traditional choices.

The Wikipedia article has over a dozen references, including a number of online books.

https://en.wikipedia.org/wiki/Model_theory