Energy conservation can take you a long way:

E = K + U = ½ kA² = ½ m v_max²

Use that if you know something about position or speed and you want to find the other.

For time use x(t) = A cos(ωt)

First determine ω, the angular frequency.

The velocity and acceleration are:

v(t)

= -Aω sin(ωt)

so vmax= Aω

a(t)

=

dv dt

= -Aω2 cos(ωt)

so amax= Aω2

This is also useful: a = -ω² x