Many universities offer an theoretical calculus sequence (often called "honors calculus") which uses either Spivak or a similar text such as Apostol. Here are a few examples.
Calculus Theory is a rigorous treatment of single-variable calculus. The obvious next step is to continue with a similar rigorous treatment of multivariable calculus. In fact some universities, notably Harvard and Stanford, do not even bother to offer their own version of our course, figuring their incoming students have already taken Calculus AB or BC in high school. Their honors sequences start out with theoretical multivariable calculus. This is typically very hard for students even with a strong BC course who have never had experience with proofs before, and now must learn them in a very complicated setting. After our course you will be extremely well prepared for an honors-level multivariable calculus course. If you are going to a school whose honors calculus sequence starts with single-variable, you should explain your background and determine whether you should wait until the multi-variable part begins (sometimes it is only offered in the Spring semester) or whether it is better to start from the beginning even though you will be relearning material already covered in this course.
There are two Harvard courses that naturally follow this course, Math 23 and Math 25. Math 25 is considered to be the harder of the two. There is an even harder course, Math 55, which is quite famous, but it is intended for truly exceptional students who have already studied a great deal of college mathematics (and maybe some grad school level math) in high school.
The single honors sequence at Stanford starts with Math 51H, and is apparently comparable to Harvard's Math 25.