Knowing the time a ball hits the ground after throwing it up in the air and its intial postion, how would you calculate its initial velocity?

the general equation is

h = -4.9t^2 + vt + c
where h is in metres
t is in seconds
v is the initial velocity in m/s
and c is the initial height

so for your question,
you know h = 0
you know t
you know v

that leaves v as the only remaining variable.
You can thus solve for v