I cannot stress too strongly the importance of seeing a wide variety of films, on film where possible, in addition to new releases of Hollywood product. Aspiring filmmakers should open themselves to the widest possible range of influences, and should explore some areas of cinema in depth. In addition, films shown on film have particular qualities of light and tonal range and blacks that are quite different on video, and the greatest films lose a lot when transferred to video. This is especially true of older films that often depend on the particular look of celluloid. Further, current streaming and online versions of films are often in the wrong aspect ratio, so that you are missing parts of the image. I would also suggest seeing films you find of particular interest multiple times, wherever possible. Chicago offers a wide variety of excellent venues. Use them! — Fred Camper
There are two excellent ways of staying in touch with showings of older films without going to individual Web sites.
One fine resource is Cine-File, which maintains a Web site and also a weekly mailing list that you can sign up for on their main page. The writers are all volunteers, but they try to explain why they are recommending certain films. There is a page called Ocular Loci that has many links.
Every venue on the list is recommended, and the list is in approximate order of preference.
The Gene Siskel Film Center shows a wide variety of international cinema, and has excellent projection.
Doc Films at the University of Chicago is said to be the oldest student film society in the US. The theater and 35mm projectors are very good, the programming is excellent, and anyone can join; you don't have to be a University of Chicago student to participate, and if you attend meetings you can propose and vote on series. The projectionists are students, so there can be glitches, but projection is usually good. Caution: travel to the University of Chicago can be unsafe; look into it before going. The Jackson Park Express bus, #6, is perhaps safest.
The Chicago Film Society shows a variety of films, including some great ones; they often show Hollywood classics. Everything is shown on film, often in rare prints not generally available.
The University of Chicago's Film Studies Center presents a number of free events: screenings of rare films, lectures, conferences.
The Music Box Theatre shows new foreign films, some great Hollywood classics, and some other things too. The projection in the main theater is a little dim for my taste, but the theater itself is a lovingly restored old movie palace, complete with twinkling stars in the ceiling.
Facets Multimedia shows a variety of foreign films and documentaries. The main theater is small and not the most comfortable, but certainly OK.
Northwestern University's Block Cinema shows a variety of films. The theater is small but well-designed, with good projection.
Operating on a low budget as a labor of love, White Light Cinema, run by Patrick Friel, shows a variety of films, most often experimental films, that you cannot see elsewhere, with often excellent, and always interesting, selections.
Though it is for only two weeks every October, the Chicago International Film Festival offers what in some cases will be your only chance to see certain recent foreign films.
Chicago Filmmakers has, for almost four decades, been showing alternative, experimental, "avant-garde" films, and documentaries.
The Nightingale is another labor of love showing alternative, out-of-the mainstream films.
If you're wondering which films to attend and which to skip, there are many ways of deciding this, including reading writing on the films and their filmmakers in books, periodicals, and online. You should try to see films that the consensus of opinion regards as important. Also, I have my own, admittedly somewhat unusual and eccentric list of favorite filmmakers.