Question 1: Usually the next step after the two basic course that cover Newtonian mechanics, electricity and magnetism, and optics is a course in modern physics. The book I used in school was Modern Physics by Krane. To do much more in physics, you need to know a bit more math including multivariable and vector calculus, differential equations, and some linear algebra, but this book requires none of that. Since you're still in high school (I'm guessing because of the Physics B and Calculus BC), you might want to read Understanding Physics by Isaac Asimov. I say this because it doesn't require math (even though you know singe-variable calculus) and will give you a great introduction to and understanding of the stuff you will learn more thoroughly later. It goes from mechanics all the way to particle physics. A fantastic book.
Question 2: It depends on what you want to do. If you want to learn to do proofs, my suggestion would be to pick up Analysis: With an Introduction to Proof by Steven Lay. It has an excellent buildup that teaches you logic and techniques of proofs and then leads you into introductory analysis, which is the theoretical foundation of calculus. Calculus by Michael Spivak is also very good and slightly similar to Apostol, teaches you basic analysis and more theoretical calculus, and has a solutions manual as well. Apostol's writing is dry, and his books are expensive. Although I've never looked at it, I do not recommend you get the Lange book.
More advanced introductory books to analysis are Mathematical Analysis by Tom Apostol, Advanced Calculus by Creighton Buck, and Understanding Analysis by Stephen Abbott, which are alternates to the Lang book. Apostol's book is by far the more advanced and difficult, but very complete. I'd recommend the others before you go to it.
If you are wanting to just learn more math, then maybe read through the differential equations and linear algebra sections in Calculus by Tom Apostol (the book you mentioned) or in Richard Courant's Differential and Integral Calculus (this book has more physical intuition built in than Apostol and probably more fun to read). Apostol and Courant also have multivariable calculus material. You need to check though as both of the authors' books have two volumes, and I don't remember what material is in which volume.
A book that would be great to go through is Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John Hubbard. I've only read the excerpts and table of contents available online, so I don't know its exact contents, but it seems like the best, although challenging, way to learn multivariable and vector calculus. I don't recommend learning linear by itself as suggested above, as it is extremely boring, and it really helps to see how it's used as you learn it, which is how the above book will treat linear algebra.