You asked for the bare minimum.
1. Protection for you from chemicals - Usually a pair of latex gloves, bib to protect your clothes and chemical goggles.
2. Very dark place - this is needed to take film out of camera, film holder, or cassette and develop it. You can use a very dark bathroom with a few plastic trays and a timer. Or you can use a bag that closes up light tight (commonly called a changing bag) where you can take the film from the camera or cassette, put it on a reel and then put the reel in a daylight developing tank. Trays can be rocked back and forth or you can physically move the film itself for agitation. Daylight tanks are either inverted or rolled back and forth to provide agitation while the film develops.
3. Water - Used to mix chemicals, wash film that has been developed and fixed, and generally keep your tools and hands clean.
4. Chemicals -
Film developer. There are lots of options but Kodak D76 handles almost any black and white film made today and is very easy to find. It is great to start with. Rodinal is another developer that is a good beginning developer. Caffenol is also good and can be mixed from locally obtained supplies like Instant Coffee, Washing Soda and Vitamin C.
Stop Bath. This is typically used to stop the development process at the end of the developing cycle. In some rare cases it is not needed. People often use water for this but water just slows the process down, it doesn't actually stop it. If you want to reuse your fixer more than once you should use a stop bath of some kind. Vinegar works but there are other options that are probably cheaper.
Fixer. This will fix the image that has been developed so that it doesn't fade away once you expose it to light. Once the film is fixed you can wash it in full light.
Wash Aid. This is used at the end to allow the water to drain off the film quickly and not leave water spots on your film.
Distilled Water. This isn't a chemical and may not be completely necessary, but if you have any sediment or metals in your tap water it makes life much easier. I use it to mix chemicals and to do the final rinse of my film.
5. Accessories -
Fluid measuring items. I use plastic 2 cup measuring cups from Wal-Mart. I write "Chemicals" on them in magic marker so someone does not try to use them for cooking or baking.
Trays or a daylight developing tank. Your choice here, most people use the daylight tanks but I actually started by taking the film off the spool and dumping it into the chemicals mixed in the measuring cups themselves. I just swirled it around with my fingers (with gloves on) and would grab each end of the roll of film once in a while to pull it back and forth through the solution. I seen it demonstrated in my little Kodak booklet. It was primitive but it did work for both 35mm and 120 roll film.
Large Plastic Mixing Bowl to dump used chemicals and contaminated wash water into rather than down the drain. If you are on a city sewer system this may not be necessary.
Clothes pins and an indoor clothes line over the bathtub are handy for hanging film. I used
Bulldog or Binder clips to weigh down the other end of the strip of film. I still do. You will need
a timer, preferably one that you can reset in the dark. Mine has illuminated numbers and they don't seem to bother the film. I buy cheap ones at Wal-Mart because they die on me once in a while. Resetting them with wet fingers does that.
A One Gallon Water Pitcher is not necessary but it stops me from running water continually and I can have a water pitcher with warm (20C) water available.
Thermometer. Black and white film is typically developed best around 20 degrees C. I bought an inexpensive analogue one that shows Centigrade and Fahrenheit scales, and so far it has worked. If it is off it is off the same amount all the time so it is repeatable.
Funnels. These will help when you need to pour reusable chemicals back in their bottle.
Plastic Bottles. To hold chemicals.
Paper Towels. Lots of them.
Film Data Sheets are absolutely necessary if you want to know how the manufacturer of the film says you need to develop their film to get good, repeatable results. I print them off the internet so I have hard copies to read before I develop my film if I feel the need.
6. Sources - Freestyle Photo is a great source for a lot of this stuff and they cater to students and teachers so they will have lots of supplies and information for people just getting started. In the beginning they were my main source of supplies and information. I called them a lot. They were always good and often pointed out cheaper alternatives from my local stores that I would not of thought of. Wal-Mart and the Dollar Store were also great sources of supplies. Of course APUG and LargeFormatPhotography are also great for online information. If you are using a film where technical data sheets are not easy to get, try
www.digitaltruth.com. Their Massive Development Chart is a tremendous resource all on its own and they have lots of other aides and resources as well.
This may not seem very minimal but I was able to keep everything in a milk crate in the beginning. Of course the desire for convenience can lead us to buy lots of nice things that may, or may not, be worthwhile. As a result I need lots more room than a milk crate to keep everything now.
Good luck and have fun.
EDIT - Once you have this mastered you can start thinking about printing and developing your prints.