What should I be able to see from the graph of the wave function? I know that the square of the wave function can tell me the most probable location of the particle, but let's say I didn't square the function and I'm only looking at a graph of the wave function itself. What should I be able to tell from its graph?
My question was inspired by these graphs from Introduction to Quantum Mechanics by Griffiths:

 A: 
I know that the square of the wave function can tell me the most probable location of the particle

The term 'location' is somewhat problematic here.
For particle in a 1D well with a normalised wave function $\psi$, the Born interpretation tells us:
$$P[x_1,x_2]=\int_{x_1}^{x_2}\psi^*\psi dx$$
Or if $\psi$ is Real, then $\psi^*=\psi$, so:
$$P[x_1,x_2]=\int_{x_1}^{x_2}\psi^2 dx$$
Schematically:

This gives us the probability of finding the particle on the interval $[x_1,x_2]$. Note that when the interval goes to $0$ (because $x_1=x_2$) then $P=0$.
The probability of finding the particle in a point location is therefore always zero and this is in agreement with Heisenberg's Uncertainty Principle.  $\psi^*\psi$ is therefore better referred to as the probability density function, rather than the misleading probability. We can only calculate a probability over a specific area of space, not in a point location.
The wave function $\psi$ shows that it changes signs at the nodes (roots, i.e. zero points). This finds important applications in chemistry because atomic wave functions (atomic orbitals) can only interact additively, thereby forming molecular wave functions (molecular orbitals or bonds), if they are of the same sign.
