Determine Acceleration from Position vs. Time Graph

For a position as a function of time graph, simply find the derivative at the point which you wish to find the velocity for. Find the second derivative of the function for the acceleration. Perhaps if you uploaded an image of the particular graph in question I could be of more use.

(The acceleration will equal zero at any point where the f(t) graph changes concavity)

 

Your first assertion, that acceleration occurs only at curves is correct. Velocity is only equal to zero when it (the position function) has zero slope (ie, no motion--- straight lines and relative max/mins) and acceleration equals zero when the velocity is constant (velocity is a straight line) and when the position graph switches concavity (inflection point). It is important to note that, though velocity may =0 at some point, acceleration may not (although it can).
Here's a few rules to help you out.
When the function is concave (up) its derivative (in this case velocity) is increasing, which means that its acceleration is positive
When it's convex (concave down) its derivative is decreasing
which means that its acceleration is negative