What would you say are the prerequisites to the Analysis books? I did single-variable calculus and linear algebra in university a few years ago, but I have to admit that I'm a bit rusty.
I'd definitely recommend having taken a college level Calculus course before and you'll get more out of it if you are basically comfortable with Abstract Algebra (know what sets, groups, rings, and fields are, and be able to think fairly abstractly about operations on those kinds of objects). That said, I think you could get through the first one with just high school level pre-calc although I think it would be hard to motivate yourself if you don't know enough calculus to see where it's heading. It does an absolutely brilliant job of starting with Peano's axioms to define the natural numbers, using those to define integers, using integers to define the rationals, introducing Cauchy sequences and using them to define the Reals, then introducing limits, continuity, etc. and Riemann integrals. That whole part is pretty much self contained with each concept rigorously (but clearly) built out of the previous ones. Some basic algebra and the ability to follow a mathematical argument are pretty much all you need. As it gets into the second volume, it expects some familiarity with logarithms and starts building into more advanced Calculus. It's still fairly self-contained but you might struggle if you don't remember, eg, what integration by parts looks like.
You can check the analysis courses offered where you took calc and linear algebra to see if you have sufficient prereqs.
Depending on the level of comfort or exposure you've previously had to proofs, it could be good to have a book covering introduction to proof handy. A book on counterexamples in analysis is also something I've seen recommended.
For proofs, something that covers direct proof, proof by contrapositive, proof by contradiction, and mathematical induction are good to be familiar with.
Delta-epsilon proofs are also good to have alternative or more accessible explanations to draw on.
If you're covering topics in multivariable calculus, then it might also be handy to cover some of the calculations from a calc 3/4 book to see implementations of it.
A good intro real analysis book will have no prerequisites, really! I learned from Ross, "Elementary Analysis: The Theory of Calculus", which starts with sequences of natural numbers.
(I'm of the opinion that the usual Calc1+2+3 sequence should be scrapped in favor of everyone taking "Advanced Calculus" first. It should probably even be taught in 10th/11th grade in place of the dreadful "pre-calc" courses many schools have. Calculus didn't click for me until my first proof-based course. That's also when I learned that a "proof" is just a detailed explanation of exactly why something is true.)