I am not good at definitions with Math (I like doing with numbers) but here is what I can say.
One of the uses of differentiation is to find the first derivative of a function. Then one of the meanings of this first derivative is its value is the gradient or slope of the tangent line to the curve of the function.
This first derivative may not be linear always (it is linear only if the function is of order 2). As you said, sometimes dy/dx is quadratic (if the function is of order 3) or polynomial.
"...rather than a straight line"
Uh, the dy/dx is not the equation of the tangent line. Rather, it is the expression of the value of the gradient of the tangent line only. So dy/dx can be quadratic or polynomial.
To get the equation of the tangent line, you need the value of dy/dx at the
point of tangency, and at least the point of tangency, and use the point-slope form of a line.
The second derivative is the rate of change of the first derivative. It is the rate of change of the slope of the tangent lines.
Zeez, the books can explain more or better.
Third derivative? Rate of change of the 2nd derivative.
Later on in your studies, you'd understand why these derivatives. Relax.