Keep a file with notes about books. Start with Spivak’s “Calculus” (do most of the exercises at least in outline) and Polya’s “How to Solve It”, to get a feeling of how to understand a topic using proofs, a skill necessary to properly study texts that don’t have exceptionally well-designed problem sets. (Courant&Robbins’s “What Is Mathematics?” can warm you up if Spivak feels too dry.)
Given a good text such as Munkres’s “Topology”, search for anything that could be considered a prerequisite or an easier alternative first. For example, starting from Spivak’s “Calculus”, Munkres’s “Topology” could be preceded by Strang’s “Linear Algebra and Its Applications”, Hubbard&Hubbard’s “Vector Calculus”, Pugh’s “Real Mathematical Analysis”, Needham’s “Visual Complex Analysis”, Mendelson’s “Introduction to Topology” and Axler’s “Linear Algebra Done Right”. But then there are other great books that would help to appreciate Munkres’s “Topology”, such as Flegg’s “From Geometry to Topology”, Stillwell’s “Geometry of Surfaces”, Reid&Szendrői’s “Geometry and Topology”, Vickers’s “Topology via Logic” and Armstrong’s “Basic Topology”, whose reading would benefit from other prerequisites (in algebra, geometry and category theory) not strictly needed for “Topology”. This is a downside of a narrow focus on a few harder books: it leaves the subject dry. (See also this comment.)
Keep a file with notes about books. Start with Spivak’s “Calculus” (do most of the exercises at least in outline) and Polya’s “How to Solve It”, to get a feeling of how to understand a topic using proofs, a skill necessary to properly study texts that don’t have exceptionally well-designed problem sets. (Courant&Robbins’s “What Is Mathematics?” can warm you up if Spivak feels too dry.)
Given a good text such as Munkres’s “Topology”, search for anything that could be considered a prerequisite or an easier alternative first. For example, starting from Spivak’s “Calculus”, Munkres’s “Topology” could be preceded by Strang’s “Linear Algebra and Its Applications”, Hubbard&Hubbard’s “Vector Calculus”, Pugh’s “Real Mathematical Analysis”, Needham’s “Visual Complex Analysis”, Mendelson’s “Introduction to Topology” and Axler’s “Linear Algebra Done Right”. But then there are other great books that would help to appreciate Munkres’s “Topology”, such as Flegg’s “From Geometry to Topology”, Stillwell’s “Geometry of Surfaces”, Reid&Szendrői’s “Geometry and Topology”, Vickers’s “Topology via Logic” and Armstrong’s “Basic Topology”, whose reading would benefit from other prerequisites (in algebra, geometry and category theory) not strictly needed for “Topology”. This is a downside of a narrow focus on a few harder books: it leaves the subject dry. (See also this comment.)