Looking for a Calculus Textbook

As I said, for a rigorous and theoretical approach to calculus, Apostol's Calculus Vols. $1$ and $2$ are very good. Depending on your background, for multivariable calculus, Spivak's Calculus on manifolds is also good. Spivak's Calculus (which does single variable calculus) is also one of my favorites.

I think I first learned calculus from Richard Courant's Introduction to Calculus and Analysis. I think Courant's and Robbin's What is Mathematics? also has good intuitive explanations of differentiation and integration.

For a book more intuitive, and perhaps something that a third grader would have background for, try Silvanus Thompson's and Martin Gardner's Calculus Made Easy.

Of course, a book that worked for me might not work for you. I would suggest that you go to a library and browse through a number of different calculus books (there are a lot of them out there), till you find the one that appeals to you the most. If you really are in the third grade, then I would assume there is no real hurry for you to master calculus, and if there is, then the books above are a good place to start.

Differential and Integral Calculus, Vol. I [Paperback] Piskunov (Author)

Try this cover to cover and if you finish this you will know more one variable calculus than you will need.

I think one's first exposure to calculus-no matter how gifted or ambitious the student is-should be a physically and geometrically motivated approach that illustrates most of important applications of calculus. Sadly,many people think that means a "pencil-pushing" or "cookbook" approach where things are done sloppily and with no careful explanation of underlying theory. That's simply not true. You can certainly do calculus non-rigorously while still doing it carefully enough to give students the broad picture of the underlying theory.

The best example of this kind of book,to me, is Gilbert Strang's Calculus. Strang's emphasis is clearly on applications and it has more applications then just about any other calculus text-including many kinds of differential equations in physics(mechanics),chemistry( first and second order kinetics),biology (modeling heart rythum) and economics and a basic introduction to probability.But Strang doesn't avoid a proof when it's called for and the book has many pictures to soften the blows of these careful proofs. This would be my first choice for a high school student just starting out with calculus.