Im still taking Integral Calculus now, but for me, if you dont know Differential Calculus you will not know Integral Calculus, because Integral Calculus need Differential. So, as an answer to that question, ITS FAIR
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Differential calculus is concerned with finding the slope of a curve at different points. Integral calculus is concerned with finding the area under a curve.
Analysis is a broader term for Calculus and the theorems behind it. It is studied both with real and complex numbers as real and complex analysis. Usually calculus just deals with the basic problems of differential calculus and integral calculus.
You must have a strong basis in Algebra, Algebra II, Geometry and Trigonometry. Usually high schools offer a pre-Calculus course which is somewhat of a conglomeration of the aforementioned courses. Then you would move into differential calculus, integral calculus, vector (multi-variable) calculus, and finally differential equations, which is considered to be at the top of the hierarchy of the calculus courses. So take Algebra, Algebra II, Geometry and Trigonometry to get your strong foundation before begining the calculus sequence.
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Once you've completed differential and integral calculus, multivariable calculus is often next step, and beyond that there is advanced calculus which generalizes calc to multidimensional spaces and uses vector-valued functions. Often concurrent with high level calculus in college courses is linear algebra and differential equations. There's nothing really 'after' calculus, because any topic in mathematics has a myriad of problems, theories, and potential applications to be explored. Calculus is, however, normally the highest level of math taught in US high schools and is a basic required course for any science/engineering major in college.