Preparing for proof-based mathematics at university

For an introduction to proofs you can look at any introductory textbook on (Naive) Set Theory or Elementary Logic. They'll teach you operations defined on sets, the various problems associated with naive set theory, a look at logical quantifiers and negation and methods of proof like contradiction, contraposition and induction. Books like "How to Prove it" by Velleman and "Book of Proof" by Hammack seem to be commonly recommended texts as well.

If you want to get a look at proof-based calculus or linear algebra then "Calculus" by Spivak will be more than enough for the former, and something like "Finite Dimensional Vector Spaces" by Halmos or "Linear Algebra Done Right" by Axler will be great for the latter.

If you're taking analysis in the first year then I'd recommend reading "Understanding Analysis" by Abbott. It's an excellent textbook although I'm using it just as a supplement at the moment.

What AS/A-Level maths and further maths topics do I need to know well before moving on to books such as "How to Prove It" and "Understanding Analysis"?

What order should I use all of those 4 books in? Should I read "How to Prove It" and "Understanding Analysis" before moving on to "Calculus" by Spivak and "Linear Algebra Done Right'?

Once I've finished AS/A-level mathematics and further mathematics, how can I prepare for proof-based mathematics at university? I will be doing calculus and linear algebra in first year.

You'll definitely need to know all of C1-C4, and I'd say most if not all of the Further Pure stuff simply because you need the mathematical maturity to read these books. You really aren't going to require the knowledge of knowing how to solve differential equations, but you need to be able to think mathematically. If you've done any BMO or STEP papers, those are good too.

"How to Prove it" is probably the most basic book in that list, and it'll teach you about the logic and set theory I was talking about. Between "Understanding Analysis" and "Calculus", I'd say sometimes "Calculus" is the more difficult book (in terms of the problems) but "Understanding Analysis" deals with a more rigorous treatment of single variable calculus (real analysis), so it really depends upon what kind of course you'll be taking.
And you can read through "Linear Algebra Done Right" whenever, although the abstract nature of the text (it does not deal with applications, nor does it even deal with determinants) might throw you off a bit.

Martin Liebeck: A concise introduction to pure mathematics

What other maths courses are you doing because analysis is the real proof based one, calculus sounds applied and linear algebra is in the middle.

Also, you would probably do an introduction to proofs course like I think mine was called 'numbers and sets' if i remember correctly.

Could you tell me what you mean by this?

are you planning to study mathematics in its own right or another subject like engineering or physics?

Engineering. Is it better for me to use proof-based textbooks or skill-based textbooks (which focus more on applying the knowledge)?

Engineering will not have any (serious) proof based maths. Just make sure you are comfortable with A level+FM material

Is there any benefit for an engineering student to go through proof-based mathematics?

well there is a benefit in anyone learning about proof and logic. But for engineering you certainly will not need to know/use proof. In fact I am certain you do not cover any serious proofs at all.

I am the type of person who really needs to understand things in order to remember anything which is why I thought focusing on proofs would be a better idea than skill-based mathematics. What do you think?

Also, would you say that GCSE mathematics, AS/A-Level mathematics and further mathematics are more proof-based or skill-based?

Is there any benefit for an engineering student to go through proof-based mathematics?

Not really, no, I think there are far more useful things to spend your time doing to compliment your studies or help with your employability, but as an endeavour it's more productive than lots of other things you could be doing, e.g. wasting time on the internet.