If you are very good at algebra, that is all you need to start. Some parts of Calculus use the "transcendental" functions, [math]e^x[/math], ln(x), sin(x), cos(x), etc. but the basic concepts do not require them. It would help if you had some knowledge of infinite sequences and series and limits of sequences and functions but any good Calculus text will at least review those in the first chapters.
I teach Calculus I and II.
You need to be very good with Algebra to begin with.
Then you will need .....
Polynomial Long Division
Trigonometric Identities [the basic Pythagoreans]
Familiarity with Exponentials and their manipulation
Complete the Perfect Square Method
Adding rational polynomials [the reverse of Partial Fraction Decomposition]
Im an undergrad in my junior year studying mathematics and comp sci. I have finished all calc I-III and I can say that you should know you algebra well and trig. Trig identities become pretty important and when and if you proceed to applied calculus you will find that trig is quite handy. Sidenote - Calc II is going to be the toughest. Not because it is necessarily more difficult, but because it pretty much entails EVERYTHING you will learn in beginning calculus.