Get a camera
Step one is get a film camera. This is easy; ask your relative/friend who never throws anything away for their old camera. If you have no friends or relatives, go to eBay and type in "35mm film camera". Buy any working camera. If it has an "auto" mode, all the better. Make sure it comes with a lens, since sometimes they are detachable. It doesn't have to be expensive; you will find plenty of results in the $20 range, and these will be fine.An Olympus XA: a cheap, powerful, always-automatic film camera.
Get some film
You can purchase new film online from any camera store, and through Amazon. If you want to play around with expired or strange films, eBay and Lomography are where you should look, but I'd recommend leaving those for your second roll of film. You can also purchase film from local camera stores, or if you're in a bind some pharmacies still stock the cheapest films. I've also met up with people selling film stock through Craigslist, but meeting internet strangers is scary.
The film you're looking for is 35mm film. There are many different types, such as color, black and white, slide film, and so on, but all 35mm film works in all 35mm film cameras. If you have no idea what to get, buy Kodak Gold or Fujifilm Superia (a.k.a. Super HQ). These are amazingly forgiving films and will give you workable photos even if you completely botch up the camera settings.
Ok, now shoot your film in your camera
Set your camera to "auto" mode and shoot your film. Don't forget to have fun! I'm serious, cameras all have built-in fun detectors and if you're not generating a sufficiently strong "fun field" then the camera automatically makes your photos turn out worse.
If the camera does not have an "auto" mode, you can learn to work manually, which is far less daunting than it seems. Auto mode was invented to help in tricky situations, but if you're taking photos outside during the daytime, then your camera settings will be as simple as they come.
Develop that film
You can set up a home chemistry kit for developing your own film, but for those of us that prize the cleanliness of our bathrooms, we send our film out to labs to be developed. Where I live in Northern California, it is hard to find good developing. There's one Costco in the area that will do it, but they'll do a terrible job. There's two camera stores that will do it, but they'll charge a ton of money and do a mediocre job. There's a lab in Southern California that I mail my film to, which ain't cheap either, but they do immaculate work. There are probably other labs that can do great work, but I've stopped looking.
Enjoy your photos
Now the nice thing about modern film labs, is that they can also scan your negatives at better quality than you can. Home scanning is not that difficult, but when you're starting out, life is much simpler just to have the lab scan them. The lab will send you digital files, just like shooting a digital camera.
Digital files are great and all, but since you are shooting film, you should also order a set of prints. Any lab worth a damn can make prints directly from you negatives, which means you'll see colors and tones in your images that sometimes a computer file will clip out.
It's that simple
Getting started with film photography is really just that simple. We live in a time where you can get a working, high-end film camera for 1/10th the original cost, and we still have access to new film and labs that will develop it. Do not be intimidated, like I once was. And if you have any questions or fears, remember that there's plenty of people just like me, who love sharing their knowledge with people. Read articles, ask questions, have fun, make photos.