For Algebra, I'd rather recommend Aluffi's "Algebra Chapter 0". The prerequisites are maybe a passing knowledge of some algebra, but mostly start with fundamentals and develop the concepts through explanation and then interesting exercises. It's a joy to work through, but can occasionally be tough.
For Analysis, I'd rather recommend Tao's "Analysis I". Tao has the style that I really enjoy where he uses a mix of intuition and light mathematical maturity to build basic Analysis from first principles. Not everyone benefits from this approach, but if you're the person who enjoys discussing and debating with the author as you develop the fundamentals, then Tao's Analysis is fantastic.
Aluffi's book switches to the language of category theory way too early. The best introductory yet "serious" book on abstract algebra is A first course by Fraleigh; for topology, the book by Munkres is at the same level.
I preferred that approach, personally. I felt the categorical treatment led to a bit more unified insight rather than treading each of the cases as independent with similarities we just "take note".
For Analysis, I'd rather recommend Tao's "Analysis I". Tao has the style that I really enjoy where he uses a mix of intuition and light mathematical maturity to build basic Analysis from first principles. Not everyone benefits from this approach, but if you're the person who enjoys discussing and debating with the author as you develop the fundamentals, then Tao's Analysis is fantastic.