Find the general solution of the equation of motion. How do we find the general solution of:
$mu$''+$ku$=0
This is the equation of motion with a damping coefficient of 0.
The characteristic equation is $m$r$^2$+$k$=0.
From here, how do we find the complex roots and get it to look like the following:
$u(t)$=$A$$cos$${w_0}$$t$+$B$$sin$${w_0}$$t$
Any help is greatly appreciated.

This is the Harmonic Oscillator equation, possibly one of the most common in theoretical physics.
So lets put your equation in  the form
$$ u'' + \frac{k}{m} u = 0 $$
and let 
$$ \frac{k}{m} = \omega^2 $$
for ease.
As you assumedly did to get that correct characteristic equation, we take a general solution
$$ u(t) = A\exp(rt) $$
giving you
$$r = i\omega$$
Hence by ODE theory, a general solution is given by
$$u(t) = A \cos(\omega t) + B \sin(\omega t)$$
as you correctly had.
Does this help?
