Calculus question increase/decrease, local max/min, concavity

I have a test soon so please tell me if I have this right.
To determine intervals of increase/decrease, find f ' (x) and it's critical points; then do the sign chart, plug those random numbers to the right and left of those critical point(s) while plugging them inside f ' (x).
To determine the point at which there is a local maximum or minimum you plug the critical points you got from f ' (x) and plug them back into the original equation f to get the height (the y value ). Now that you have the point you could determine if it was a local max or min by the sign chart you drew for f ' (x) and if it is + on the left of that critical point and - to the right of that c. point it is a maximum; vice versa.
To determine the concavity you find the 2nd derivative f '' (x) and you can use the critical points from f ' OR you can find the critical points for f '' (x) and then you plug the critical points ( whichever option you choose) back into f '' ( x) to determine the concavity. Using the sign chart again.

Correct me if I'm wrong !

I think you have it all correct. It can be summarised as

f '(x) > 0 ----> function increasing.
f '(x) = 0 ----> turning point
f '(x) < 0 ----> function decreasing

f "(x) > 0 ----> function concave up
f "(x) < 0 ----> function concave down

f "(x) = 0 and f '(x) =/= 0 ----> point of inflection
f "(x) = 0 and f '(x) = 0 ----> usually a horizontal point of inflection but there are rare cases where it is a local maximum or minimum.

You are absolutely right.