Do you need calculus to learn pure mathematics?

[Jjasongray]

Would an intro to proof book be the best thing to start with then?

Spivak is kind of an intro to proofs book. I would recommend starting with it. It doesn't assume you know anything at all and it sort of eases into it. Once you start limits there's a bit of a difficulty spike but once you start "getting it" the rest comes easily since limits are the hardest topic in the book and the one that everything else is based off in the subject. I took a class with it having no knowledge of proofs at all and it turned out okay.

Also, don't ignore calculus! It's really a pretty interesting subject. While I admit differentiation is very dry, integration is an art. Differential equations are also really fun in my opinion.

It's also insanely powerful in any applied subject. When I first learned calc I was mystified it worked so well! I would do problems and would wonder how the hell such simple techniques got answers to seemingly complex questions.

That's usually how calculus is taught at first, with a focus on computation and almost no proofs. That's why Spivak is a recommendation on here because it pretty much does the opposite.