I almost did a film minor as an undergraduate. What stopped me was that I would have had to take art history courses as part of the film minor. Given the amount of time that I have spent in art museums since, this was a stupid decision, and one I quickly corrected as a graduate student.
From about 1979 to 1985, I saw an average of three films a week. While living in and near Toronto, I used to attend the Toronto International Film Festival (or "Festival of Festivals", as it was then called), seeing up to twenty-five films during the week and two weekends of the festival. More than that, and they start to all blur together.
There's no shortage of top-10 or top-100 film lists on the Internet. Here are five somewhat more obscure films I would recommend.