Before the internet people who wanted to be mathematicians had to read books, so books were extremely important. I was reading Littlewood's "A Mathematician's Miscellany" where he gives the sequence of books he read before university (see pg.67 here: https://archive.org/stream/mathematiciansmi033496mbp#page/n77/mode/2up )
In that spirit, which sequence of books would you recommend to someone having only a high school background of math but who is willing to learn?
Also what do you think of the following sequence? :
Apostol's Calculus volumes
Artin's Algebra
Rudin's Principles of Analysis
Axler's Linear Algebra Done Right
Marsden/Hoffman's Complex Analysis
You're list is fine. Only thing is that using several books to cover different topics can sometimes be difficult, as you sort of get used to a particular Author's technique and notation. But at some point you will have to read from different authors so I guess it's up to you.
How about finding a book that covers all topics at undergraduate level. I recommend the one I used: Mary L Boas. Mathematical Methods in the Physical Sciences 3rd Edition. This book covers everything you mentioned from Linear Algebra to Complex Analysis, Calculus of Variations and much more, in nice detail might I add.