We caught this old (1932!) movie on tv and it was wonderful, which led to deciding to watch more good classic movies, which led to finding imdb's list of top movies, and, wouldn't you know, there's an app for that:
It's not that fancy, but does list the movies and let you check them off and sort them as you watch. It's called MovieList and I'm pretty sure it was free, but I can't find it on the internet anywhere which is weird.
They just don't make movies like they used to.