You should not think of the Schrödinger equation as a true wave equation. In electricity and magnetism, the wave equation is typically written as
with two temporal and two spatial derivatives. In particular, it puts time and space on 'equal footing', in other words, the equation is invariant under the Lorentz transformations of special relativity. The one-dimensional time-dependent Schrödinger equation for a free particle is
which has one temporal derivative but two spatial derivatives, and so it is not Lorentz invariant (but it is Galilean invariant). For a conservative potential, we usually add to the right hand side.
Now, you can solve the Schrödinger equation is various situations, with potentials and boundary conditions, just like any other differential equation. You in general will solve for a complex (analytic) solution : quantum mechanics demands complex functions, whereas in the (classical, E&M) wave equation complex solutions are simply shorthand for real ones. Moreover, due to the probabilistic interpretation of , we make the demand that all solutions must be normalized such that . We're allowed to do that because it's linear (think 'linear' as in linear algebra), it just restricts the number of solutions you can have. This requirements, plus linearity, gives you the following properties:
1. You can put any into Schrödinger's equation (as long as it is normalized and 'nice'), and the time-dependence in the equation will predict how that state evolves.
2.If is a solution to a linear equation, is also a solution for some (complex) . However, we say all such states are 'the same', and anyway we only accept normalized solutions (). We say that solutions like , and more generally , represent the same physical state.
3. Some special solutions are eigenstates of the right-hand-side of the time-dependent Schrödinger equation, and therefore they can be written as
and it can be shown that these solutions have the particular time dependence . As you may know from linear algebra, the eigenstates decomposition is very useful. Physically, these solutions are 'energy eigenstates' and represent states of constant energy.
4. If and are solutions, so is , as long as to keep the solution normalized. This is what we call a 'superposition'. A very important component here is that there are many ways to 'add' two solutions with equal weights: are solutions for all angles , hence we can combine states with plus or minus signs. This turns out to be critical in many quantum phenomena, especially interference phenomena such as Rabi and Ramsey oscillations that you'll surely learn about in a quantum computing class.
Now, the connection to physics.
1. If is a solution to the Schrödinger's equation at position and time , then the probability of finding the particle in a specific region can be found by integrating around that region. For that reason, we identify as the probability solution for the particle.
We expect the probability of finding a particle somewhere at any particular time t. The Schrödinger equation has the (essential) property that if at a given time, then the property holds at all times. In other words, the Schrödinger equation conserves probability. This implies that there exists a continuity equation.
2. If you want to know the mean value of an observable A at a given time just integrate
where is the linear operator associated to the observable. In the position representation, the position operator is , and the momentum operator, , which is a differential operator.
The connection to de Broglie is best thought of as historical. It's related to how Schrödinger figured out the equation, but don't look for a rigorous connection. As for the Hamiltonian, that's a very useful concept from classical mechanics. In this case, the Hamiltonian is a measure of the total energy of the system and is defined classically as . In many classical systems it's a conserved quantity. also lets you calculate classical equations of motion in terms of position and momentum. One big jump to quantum mechanics is that position and momentum are linked, so knowing 'everything' about the position (the wavefunction at one point in time tells you 'everything' about momentum and evolution. In classical mechanics, that's not enough information, you must know both a particle's position and momentum to predict its future motion.