Developing film is a rewarding, easy, and forgiving process: I used to do it with young children at summer camp and it never failed (even in 85 degree weather!).
At a minimum you need a developing tank, developer, and fixer.
It helps also to use a stop bath (a mild acid, essentially dilute vinegar but purified).
You also temporarily need a perfectly dark room, like a closet at night. (I once used a basement at night with aluminum foil taped over the windows. Check the room out by sitting in it for 5-15 minutes with your eyes open: if you still can't see even a glimmer, you're fine.) If you plan to do this often, get a changing bag: the film and developing tank go into it along with your hands. Inside this lightproof bag you open the film canister (using a bottle opener), load the film in the developing tank's reel, place the reel in the tank, close the tank, and you're good to go: everything else happens in the light.
For mixing the chemicals you will want a reasonably accurate way to measure liquids. And light-proof plastic or glass bottles to store them.
Finally--this is essential--you need a thermometer that is accurate in the range 65-75 degrees F (18 - 24 degrees C).
The basic process consists of loading the film into the tank (practice in the light with a roll of old exposed film), pouring developer into the tank, agitating the tank on a specified schedule ("agitation" means inverting slowly once every 30 or 60 seconds), emptying the developer, pouring in the stop bath (or water), emptying that, pouring in the fixer, emptying that after a period of time, opening the tank and rinsing the now-developed film with water (under a tap usually does fine), and hanging the film in a dust-free place to dry (I used to clip the film to the shower rod in the bathroom and weight the lower end with a couple of clothespins). Afterwards you cut up the negatives and place them in glassine or plastic sleeves for protection. Then you're all ready to make a contact sheet and proceed to enlarge them...but that's a topic for another day.
Instructions for development time and agitation usually come with the developer, which often is a packet of powder you mix with warm water and let cool. The time usually is from 2 to 8 minutes, depending on temperature (whence the need for the thermometer); the whole process takes about 30 minutes plus drying time.
Make sure you rinse everything thoroughly and clean up carefully when you're done: these chemicals don't do good things to the bathroom or kitchen furniture if left lying around. And if you store the chemicals in a public place--even temporarily--please label the bottles clearly and carefully. When I was a kid and first learning how to do this I put the warm, newly mixed fixer into the refrigerator to cool. It was in a recycled tonic water bottle, not too well labeled with a magic marker. My mother made a gin-and-fixer cocktail with it and almost drank it...