On numerics: there are entire textbooks at a relatively elementary level (no differential geometry) with titles like "Simulating Hamiltonian dynamics". At the end of the day, verifying that a discrete map (approximating a Hamiltonian system) is symplectomorphic is a matter of linear algebra.

On theory: there are a couple of good lectures by Dusa McDuff on YouTube; she also has a short tutorial-like PDF somewhere googleable.