I've had success teaching an elementary school student about binary numbers. I listed powers of two up to 512 and claimed I could make any number up to 1000 by adding the powers of two, using each one at most once. I had the student give me numbers and demonstrated how to make them out of the powers of two.
Then I had the student do it herself for some small numbers using guess-and-check. Then we made a table with powers of two as columns and sequential integers, from 0, as rows, and put an X in any entry if the power of two for that column was used to make the number in that row.
We looked for patterns in the X's, and were introduced to binary numbers that way. By the end of two hours, she could convert any number between binary and base ten, and could add and multiply in binary.
I haven't tried to do it and don't know how, but I suspect you can introduce basic group theory to students of this age, too.
Have you considered taking a class in "Advanced Calculus" or an (undergraduate) introductory class in mathematical anaysis?
It is often the case that Calculus I, II, III and Differential Equations are taught as "rules to apply" and "procedures to follow", so it is not unusual for students to feel as you do.
The "theoretical" understanding of Calculus is usually the goal of courses in Advanced Calculus and/or undergraduate mathematical analysis. If such a class is offered at your university or college, you might want to consider enrolling.
If you want to cover this territory on your own, I'd suggest previewing Serge Lang's Undergraduate Analysis, to see the topics covered, and if it looks "doable", obtaining it (perhaps you can find it at your campus library). There is an accompanying text: Problems and Solutions to Undergraduate Analysis, which I'd recommend, for self-study.
At the same time, try to obtain or borrow a copy of Walter Rudin's PMA: Principles of Mathematical Analysis (which has excellent exercises). You can obtain Prof. Silvia's COMPANION NOTES: A Working Excursion to Accompany Baby Rudin, to help work through the text on your own.
Here's another "self-study" option: See MIT's OCW Introduction to Analysis I. The text for the class is Rudin's PMA. Through the MIT site, there are problem sets to work through, lecture notes to accompany the text, and exams/quizzes available to test your knowledge. The prerequisites listed for the class are multivariate calculus and differential equations.
Note: If you find the above to be overwhelming, you might want to revisit Calculus with an emphasis on theory and concepts. For example, MIT has a course: Calculus with Theory, which uses the text by Tom Apostol: Calculus I, and a follow-up class: Multivariate-Calculus with Theory, which uses Apostol's Calculus I and II. The classes, together, cover calculus, differential equations, and introductory linear algebra, so you could follow the syllabi, lecture notes, work the problem sets, and in so doing, revisit most of the material you covered, but at a deeper level.
Best Answer
If you haven't been using "The Art and Craft of Problem Solving" by Zeitz, you should. Like, right at this moment. :)
For neat problems... you might want to look into "Which Way Did The Bicycle Go?" by Konhauser, Wagon and Velleman.