EDIT to be perfectly clear
You should not learn to shoot film.
You should instead ask yourself the question before every shot:
"Will anyone care about this photo?"
The answer is, for the most part, no. No one cares about your photos, just as no one cares about mine. You have to make them care, and you make them care by taking shots that resonate with the viewer. Some viewers will continue not to care about what you do, while others may find great appeal. Part of the mystery of creating art is accomplishing this feat. Technology enables you to do it, technology informs the decisions done when doing it, but ultimately, the resonance comes from your own vision of the world and your ability to make your technology realize your vision.
All of the arguments by the other posters basically boil down to "Slow down, digital allows you to move to quickly without thinking about the consequences of your actions." And to a certain extent, that's true. But consider the drawbacks of film:
- Film results are not instant. When you experiment with different apertures, isos, shutter speeds with digital, you get instant feedback via review. With film, you have to wait until you get the results back, and ISO is not an instant change.
- Film processing can only be done once. You get one negative. You blow the negative, you've lost the shot. You process it one way, hard to get it back to another.
- Film processing is expensive if you do it yourself. Chemicals, knowledge, equipment, and space may or may not be cheap for you, but they are all part of the package of learning to shoot film completely, to get a final print. With digital, you need a computer, some programs, and a printer (granted, making a print does require some knowledge of calibration, and that wasn't so hard for me). As film chemicals become harder to find, as well as the space and plumbing for a dedicated darkroom, digital gets to be much cheaper, faster.
- Film processing is expensive if someone else does it for you, and you may not like the results. If you go to the guys down the street at the corner drug store, you'll have to settle for 'auto processed' colors. If you go with a lab, expect to pay a lot.
- It is very debatable if film is of 'higher quality'. I would think that, now, it's not really a debate, but some people are willing to shoot brick walls until the cows come home to prove one point or another. I leave that debate to them.
I think that the best think you can do to achieve the same results everyone else was talking about (thinking about composition, locking yourself to aperture/shutter/iso, etc) is to get a prime lens and a little bit of discipline and thinking about the shot you want to take. Getting involved with all of the technical difficulties of film is just a more expensive way of doing the same thing.
Meta: I already have two downvotes on this answer, but I think that this will become one of the most asked and debated questions on this site. I also think that it will border on the religious. Let me not mince words, because I'd rather lose rep points now to answering the question (or at least having a series of responses to point to), so that repeated questions can just go here.
Previous Answer (2 downvotes):
For me, the question is whether or not you're doing this as a hobby, or professionally. As someone who has done pro event photography, the downsides of film are just too great to shoot with it compared to digital. Consider that with digital, when I'm taking a shot of the bride, the groom, seven bridesmaids, seven groomsmen, and a two-year-old ringbearer, spamming that shot is critical. Someone always looks away at the worst moment, and so getting a three to five shot burst means I don't have to worry about it. I can also chimp through those shots and make sure that I got something reasonable.
Bursting is just necessary for the first kiss, the handoff, etc. Those things don't come twice, and trying to one-shot-one-kill sniper that kind of thing means that I end up hoping my second got the shot. Of course, timing plays a role, and it's a big one, but the milliseconds between facial expressions can tell different stories.
Now, if you're doing this to learn, as a hobby, then that's a different story. Learning film is like doing any other exercise to take you out of your comfort zone, forcing you to know what you're doing (and maybe even writing a log of shots you've taken, so that when you get the roll back, you'll know what you did).
- Digital lets you get sloppy with metering, especially with a RAW capture of a huge dynamic range that can be fixed in post processing using fill light/exposure correction/recovery/etc.
- Digital lets you get sloppy with cropping. Off by a little bit, no problem. Considering using your 60mm lens as a 120? Lots of sensor lost, pretty sloppy.
- Digital lets you burst, meaning that if you want to get that perfect Doisneau moment, you either hire models (which he sometimes did) or get very very lucky with timing, and that luck turns into skill the more you do it.
With digital, you still need to:
- Compose. I take issue with the idea that digital shooters don't slow down and compose their shots. I'd say that bad digital shooters don't slow down and compose their shots, just like bad film shooters don't slow down and compose their shots. If you don't know the rule of thirds (and when and why to break it), then switching to film doesn't magically give you this.
- Learn the etiquette of candid/street photography. Film or digital, people just don't like lenses in their faces sometimes. Switching to film doesn't naturally give a better rapport with the subject, or make you a better 'ninja shooter', grabbing candid shots that no one knows happened (ok, maybe with a soundless Leica m7 shutter, but a point-and-shoot can be soundless as well).
- Know the times of day and how that plays with light. Shooting in direct sunlight still sucks in digital, even with all the dynamic range a camera can take. Shooting at dusk, in the so-called 'magic hour', is just a good idea all around.