The best way to start with calculus is the one by Gilbert Strang, who explains everything on the first page of his book (and the rest of the book are "just" examples).
The first half of the first page shows a drawing of the speedometer and odometer of a car and it explains what they are, and that they are not independent but related in a special way. On the second half of the first page it says that differential calculus is the task of computing the speed from the distance, and integral calculus is the task of computing the distance from the speed. Then it says a lovely sentence "this is not an analogy, this is the real deal and we have already started with the subject, and this is actually all there is to it". Then in the rest of the page it explains in a couple of sentences how can you compute speeds from distances and vice-versa, and why you need a constant of integration, and so on. It also proves the fundamental theorem of calculus. The rest of the book consists in concrete examples and a few more constructions, up to Taylor series.