Running water is convenient, but not necessary. You need running water to mix the chemistry and to wash your film and prints. Both of these tasks can be done in the light so you can do them at the kitchen or bathroom sink.
For developing film you only need total darkness until you get the film on the reels and into the light proof developing tank.
For printing you will need enough room for the enlarger and at least 4 chemistry trays (dev, stop, fix, and a holding bath of water). Preferably you would be able to have the chemistry away from the enlarging area to avoid contaminating developed film and unused paper.
Once your film or prints are fixed, they are no longer light sensitive.
You will need film developer, print developer, stop bath, fixer, wash-aid, and photo flo.
You can find out about specific brands of chemistry at the manufacturer's websites such as
www.kodak.com or
www.ilford.com
There are many different options; just stick to the basics until you get it figured out. Then you can start trying different developers and techniques.
The basic Kodak film dev is D-76. Their basic paper dev is Dektol.
Indicator stop bath is nice, as it turns purple when it's no good. But some folks use straight water or water mixed with lemon juice or vinegar.
There is fixer and rapid fixer. Rapid fixer acomplishes the job in less time. Some folks like to use rapid fixer because they feel that less time in the fixer leads to a higher archival quality.
Kodak calls their wash-aid Hypo-clear. This helps wash the fixer from the prints. It is usually needed for film and fiber paper prints.
Photo flo is a Kodak brand name. I don't know what other companies call it. It helps avoid water spots and mineral deposits on drying film.
One more chemical, which is optional, but I find handy, is Hypo-check. This is used to determine if your fixer is still good. If you don't know, "hypo" is another name for fixer.
Good mail order sites for paper and film are
www.freestylesalesco.com and
www.adorama.com
I usually buy my chemistry at the local photo supply shop to avoid paying lots of shipping charges as chemistry can be heavy, but the above mentioned sites have a wide selection of chemistry too.
Instead of print trays, I use plastic tubs I get at the hardware store. They are cheaper and have higher sides to help avoid spillage.
Good luck. I set up my first darkroom in my bedroom about eight years ago. Now I am fortunate to have an entire room to dedicate to my photography.