Progressing on to university proof-based mathematics

Generally at university level they would start off by having you prove simple/trivial things to get you used to the ideas and concepts involved.

One book I read before uni was 'what is mathematics' which contains a lot of proof-based material. However, there is not as much linear algebra in there. There is some overlap between that and a typical a-level textbook but there is also differences in content.

So after doing A-Levels, do you recommend reading 'What is Mathematics' to prepare for proof-based maths? After that, what textbooks do you recommend for proof-based calculus and linear algebra?

I read that book during a-levels.

Once you're at uni you would be recommended specific textbooks for each module in the course although personally I found that the lecture notes provided contained all the content I needed to know.