Tom M. Apostol's two-volume book is "the other" classic besides Spivak. It's rigorous, but it's a rigorous calculus text, not a "baby real analysis" book like Spivak. It covers single-variable calculus, basic differential equations, elementary linear algebra, multivariable calculus, and the beginnings of probability theory. There are a large number of exercises, both computational and theoretical. Some of them are challenging.
Small points against this book are that Apostol's writing is somewhat dry and boring and that he covers integration before differentiation, which some people find strange. It's also quite expensive, but it's old and has been a standard text for many years, so your library is likely to have a copy.