For differential calculus, you need to handle derivatives of products and quotients. Next you should learn about derivatives of the basic functions - trig, exponential, log. After that you could start looking at integral calculus.
just another idea: maybe you should study related rates - a useful technique
as for integrals, you want to start with the Reimann integral. I don't know your purpose of study, but you probably don't want get too deep into the analysis - just the idea adding areas of rectangles under a curve and then making them thinner and thinner (i.e. a limit)
Note the difference between definite and indefinite integrals. The Fundamental Theorem of Calculus is, well, pretty fundamental.
Get the basics down: polynomials, trig functions, logs and exponentials
Then integration by parts. Substitution. Trig Substitution (tricky). I recall partial fractions as well...
A few basic applications might be solids of revolution, and arclength.