I'm looking for a script/ book on a very basic level. I am going to finish high school soon and go to university to study mathematics. The books I tried to read (mainly in German, so naming them won't be of any help) are very difficult to understand. I am willing to try out reading a book in English, so if you have a suggestion, please let me know!
In school, we learned about integrals and derivatives, and the Gauß-Algorithm.
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I suggest you just have fun and wait for University learn the formal stuff. Here's a related question and answers:
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I highly encourage you to study some logic if you haven't already done so in highschool.
My favourite 'first book' on the subject is Sweet Reason by Henle, Garfield, and Tymoczko. I think every student wanting to study mathematics and computer science beyond an elementary level should read this book!
The book covers sentential, predicate, modal logic, many-valued logics, and general argument and reasoning skills. It has many exercises and an accompanying website.
The book will make you get comfortable with things like quantifiers and when it comes time to do some mathematics, you can focus on the mathematics instead of the language(s) of mathematics.
Here I'll introduce some books, and (maybe) lecture notes, not oriented to promote any specialized topics in mathematics, but a necessary knowledge base that I think is good for pre-freshman in university-level mathematics.
Furthermore, I'll continuously update this post, unless it is disagreed.
General $$\textbf{ How to study for a mathematics degree }$$ Lara Alcock, $OUP, 2013$. Though the name of the book is not as interesting, it is, definitely, a good way to introduce one to a thinking style of advanced mathematics.
Analysis $$\textbf{ Analysis I }$$ Terence Tao, $Springer, (III\ Edition)2016$. A really good book in introducing the way of thinking in a constructive, based-on-axiom ways. No need further introduction, as the name of its author is enough to explain. You may also find Tao's lecture notes here at UCLA.
$$\textbf{ Mathematical Analysis I }$$ Vladimir A. Zorich $Springer, 2002$. My self-introducing book during my first year of high school, while I've finished all A-level syllabus. It gives a relatively fine way of teaching, with a level of difficulty, and also thanks to the use of logical notation by Zorich, it may give some awkwardness when first seeing it. But if you have familiarized with it, and also, trained yourself with the exercises, it must give you a better and wider view for your future studies, at least for myself.
Some other lecture notes
Calculus (Preliminary level): $Oxford$
Calculus (differential equation): $Oxford$
Calculus (vector calculus): $Cambridge$
Group theory: $Cambridge$
Tbc.