This is what you'll need, bare minimum:
2 fermenting vessels (one can double as a your bottling bucket, if you want)
A huge kettle to boil stuff in. Buy one of those turkey fryer kits at WalMart for cheap - they come with a propane burner so you can brew outside, the kettles are large enough, and they are cheaper than a plain stainless steel kettle. Some folks say aluminum is bad, others say it's A-OK.
A racking cane and tubing - this is so you can move your beer from one container to another. A bottle filler will also be needed.
A bunch of empty bottles and caps - figure 48+ 12oz bottles per 5 gallon batch.
That's the cheap way to get into brewing. If you've got a way to cool the fermenting beer to about 50F, then you can brew any type of beer you want. If not, you can brew only ales - stouts and other dark ales fit that bill perfectly.
A lot of the cheaper kits come with two food-grade plastic buckets that are 6.5 or 7 gallons each. One has a spigot at the bottom for bottling time.
I hate those kits. Reasons why are that if you scratch it, mold and bacteria will get in the cracks and ruin your beer; also, they can leech a plasticky taste into your beer if they aren't high-quality. Some people swear by them, I swear at them and use glass carboys. If you go this route, you'll spend a little more, but cleaning is easier. You'd need one 6.5 gallon carboy, and one 5 gallon carboy. If you still plan on bottling exclusively, you'll need a bottling bucket, or you can use a bottling bucket instead of the 5 gallon carboy.
As far as brewing, if you go the quick easy way, you can be sipping on your first batch of homemade beer within 3 weeks - 1 week of fermentation, then rack to your bottling bucket, prime and bottle the beer, let it bottle condition at room temperature or so for about 2 weeks so it can carbonate, and then chill a few down and drink 'em.
The key thing to remember is sanitation. Beer starts off as wort, which is not much more than sugar and water with a few proteins from grains and hops tossed in. Bacteria, wild yeast, and mold thrive in this environment. You've got to keep things clean enough for your yeast to get going with fermentation. Then, you'll have two things working in your favor - hops (which are a natural preservative) and alcohol (which kills stuff).
This post is long enough. Ask me more questions if you have any.
Check out
www.morebeer.com. That's where I order nearly all of my equipment from. Ingredients you can get at local homebrew shops.
[edit: grammar, and sepllng]
{edit of the edit: here's a brief non-comprehensive list of my beer brewing equipment:
2 6.5 gallon glass carboys
1 5 gallon glass carboy
4 5 gallon cornelius kegs
1 5lb CO2 tank
1 20lb CO2 tank
1 dual gauge CO2 regulator
1 4-way splitter for CO2
enough tubing and connectors to connect all of this up at once
racking cane
bunch of stoppers
a few cases of bottles that I don't use anymore
counterpressure bottle filler so that I can start using the bottles again after I keg the beer
1 500ml erlenmeyer flask for yeast starters
1 1000ml erlenmeyer flask for yeast starters
2 1 gallon bottles that I'll later use for mead
1 7.5 gallon stainless steel brew kettle
1 copper wort chiller
nylon bags for grains and hops
funnel with particle screen
wine thief for extracting samples
hydrometer for measuring specific gravity
thermometer for steeping grains
1 20 cubic foot chest freezer with a temperature controller to keep the temperature within a 4 degrees range
1 wing capper
1 bench capper
a bunch of different style yeasts in the fridge
a bunch of other stuff that I can't even remember
You thought guns were an expensive habit? Try getting seriously into homebrewing...]