I bought Spivak's Calculus a month or so ago, and after doing a few problems from the first chapter, it's apparent that I need some type of foundational knowledge in formal maths and proofs.
What did you study prior to Spivak? What books did you use? I've purchased Velleman's How To Prove It, but I'm not sure if this book will help me tackle an introductory elementary analysis book.
On
I think persistence and patience is the only requirement for Spivak's Calculus. But, if you are still not sure, maybe give Ross a chance. The book is intended to be an introduction to Calculus without any proof based knowledge.
What you need to read a book like Spivak for the first time is what can loosely be termed "mathematical maturity." Or at least the beginnings of it (the definition keeps getting more demanding the further your studies take you). The only way to obtain mathematical maturity is by reading books like Spivak. I do not think there is another book you need to prepare you; Spivak is one of the canonical choices for a first "rigorous" maths book.
Mostly, you just have to press ahead, keep putting in effort, and ask questions. If you put in real work on a problem and post it here you are likely to get some outstanding answers. Even if you don't understand some things completely, I would argue to keep moving forward. Many of the things you don't understand in Chapter 1 will start to solidify in your mind as you move forward. (That is not to say just skim over things... really, really work at them, but don't be afraid to continue covering new material even while the earlier chapters are still solidifying in your brain. Just don't get bogged down on one problem or section.)