Add type checking/inference to my programming language

Question

I am currently creating my own compiled programming language, and I have come to a point where I would like to start working on the type system and introduce some type checking. Ideally I would like to introduce type inference.

Right now I'm using the Visitor pattern to traverse the AST produced by the parser. I started implementing a pretty naive and ugly solution, but wasn't very happy with it. It seemed pretty hacky. Type checking that fits with this pattern would be great, but not critical.

I have looked into algorithms like the Hindley-Milner. I'm developing the programming language in Swift, so I found this GitHub gist that shows how it can be implemented, but opening the lid on this code immediately gets me to Lambda Calculus, which seems like a beast. I know Swift uses a variant of the Hindley-Milner type inference, so that would be incredibly sexy to implement, but it's way over my head at this point.

Since this is my first programming language, what's a good first attempt at a type system with type checking, possibly with type inference?