I think that an important precursor to being compassionate is the ability to sense what others might be going through and wanting to help. The fact that you were moved by the film (which I haven't watched btw) and want to be more like Mr. Rogers is an indication that the precursor exists.
Other than that, we often strive for bigger changes that we can make. Instead, we must start with our immediate circle and always keep an open eye for what we can do to help this very minute in our family, neighbourhood, work or friend circle.
Anything counts - from fixing someones computer or helping them with taxes to just catching up with someone for coffee and hearing them out about the difficult ordeal they are going through.