There are a couple fundamental ways to brew. On is with malt extract. It is a gooey molasses like syrup. The other is all grain, in which you go thru all the steps to make the malt extract. The simplest is to start with malt extract.
Like most, I started out on the stovetop. If you have a large stockpot, then that is enough to cook in. For stovetop, you usually only cook 1.5-2 gallons of the wort(unfermented sweet beer), and then add it to cool water. Really all the more equipment you would need is an inexpensive plastic fermenting container, an air lock, and a siphon tube.
I moved on to a propane burner/stand, and cooking outside/garage. It heats a lot faster, and you can boil a complete batch, which has benefits in flavor. I also moved to glass fermenters(carboys). Easier to keep clean and sanitized.
In amoungst this, I switched from 12oz bottles to champagne bottles. Much quicker to bottle the larger bottles than small. And, most times, I poured for two people.
Later, I moved to a larger gravity flow all grain system. With this system, I can brew from malt extract, all grain, or a mix of both. I primarily do 10 gallon batches.
I also, do not bottle beer. I keg everything. Much simpler and faster.
I think I posted pictures here before. I will look them up, or repost this evening.
Good thread. Can than the experienced brewers here post some info about their brewing equipment, techniques and recipes? I'd like to give the home brewing thing a try.