My sister wants me to come to her apartment and show her and her roommates how to brew a batch of beer. I am an all-grain brewer, and have spent a couple years brewing on my rig, but her request got me thinking about this exact topic.
I agree with a lot of what has been said, but I'll add a bit. It was a little unclear what the OP already has on hand, so excuse me if anything is redundant.
First of all, stick with extract and steeping grains. The ingredients will be more expensive, but the equipment requirements are so minimal that the costs will off-set should you decide not to stick with the hobby. I also do not think it is a bad idea to purchase 6 gallons of bottled spring water for use on brew day. It makes measurements easy, and you know the water is free of chlorine.
This is long, so I'll break it down the by process.
Research
Use the free tutorial at howtobrew.com to learn about extract brewing with steeping grains. It doesn't get cheaper than free. A little reading can help you avoid big mistakes, and there is no bigger waste of money and time than a stupid mistake that ruins a batch.
Brew day
Boil
The first thing you need is a 3-5 gallon pot with a lid. You will be doing a stove top partial boil. I would shoot for the 5 gallon pot.
The next thing you need is a large stainless whisk or spoon, which you should already have. This will serve to break up and dissolve the DME/LME. A quick note, use the lightest extract available, and get your character and color from the specialty grains.
Also, you will need a grain bag to hold the steeping grains, and that should be included with your kit.
Use the first 2.5-3 gallons of bottled water, and conduct the boil, uncovered, per instructions. A little jar of foam contol will do you good here.
Post boil
Cover the wort with the brew pot's lid, and submerge the kettle into a sink full of ice water. It is not the quickest way to chill things down, but it is the cheapest, and with a partial boil volume, it is good enough.
Cold side + Fermentation
This is where I really started thinking. I really like the idea of purchasing two brewing buckets, each with a ported lid, and each equipped with a bottling spigot. These buckets are the exact same, so mark one for bottling and one for fermentation. This also brings me to cleaning and sanitation. This is not the place to cut costs. A small tub of PBW, and a small jar of star-san are must haves. Do not be lured into the world of one-step cleaning. While the wort is cooling (it will probably take 45 mins or so), clean the fermentation bucket and its components with PWB, rinse with warm water, and sanitize with star-san. When that is done, rehydrate your dry yeast in 100F water, which had been pre-boiled in a pyrex container, and is now cleaned, sanitized, and covered with sanitized foil. Timing is a bit of an issue here, because you don't want to rehydrate more than 20 minutes before pitching - if you do, the yeast will start spending reserves.
Add the remaining 3 gallons of "dilution water" to the sanitized fermentation bucket (It will probably be 2-3 gallons). Now just dump your cool wort right into the fermentation bucket on top of the spring water. Dry yeast have minimal oxygen requirements, so the dumping will serve to oxygenate as well as mix the two liquids. Now pitch the rehydrated yeast, stick an airlock into the ported lid, and place in a stable, cool (65-70F), interior closet for 3 weeks.
Bottling
Three weeks later, set your fermentation bucket on a counter, and let the trub you just kicked up into solution settle. Next, measure out the appropriate amount of corn sugar and dissolve it in the microwave in a pint or two of water. While your priming solution is boiling and then cooling, get the PBW and Star-san out, and clean and sanitize your bottling bucket (and components - spigot, lid, gaskets), a 5' length of 3/8" vinyl tubing and your bottling wand. Dump your priming solution into the bottling bucket, and cover it back up with the ported lid. Now, run your 3/8" tube from the spigot on the fermentor down through the hole in the lid of your bottling bucket (which is sitting on the floor). Open the valve on the fermentor, and allow the beer to drain through - it is a closed transfer with no syphon required! This would be a good time to get your bottles cleaned and sanitized (if you have not done it already), and you can use the dishwasher racks to help you with this. Now elevate your bottling bucket onto a counter or stool, and hook the bottling wand up to it as described on these forums (I really like Revvy's preferred method, which uses a 1"-2" piece of cut tubing to make a hard connection to the bottling bucket's spigot - google it). Fill your bottles, and cap with a wing capper. Put them back into the closet, and reserve judgement for 3 weeks.
So a quick inventory
6 gallons) spring water
1) 5 gallon brew pot
1) stainless spoon or whisk
1) extract w/ steeping grains kit of choice, with a grain bag and dry yeast (US-05 is a great choice that is forgiving to fermentation temperature) A pale ale kit is ~ $30
1) small jar of foam control (optional)
1) microwave-able pyrex container (for yeast rehydration and for priming
solution)
2) brewing buckets, each with a ported lid, and a plastic spigot valve
1) air lock
5') 3/8" vinyl tubing
1) small PBW
1) small Star-san
4oz ) corn sugar for priming
1) bottling wand
1) bag of caps
1) wing capper
48-52) pry-top repurposed brown bottles
I think those instructions will maximize results with minimal investment. The keys to your success will be using a highly fermentable and fresh light extract, not underplaying cleaning and sanitizing the cold side equipment, properly handling your dry yeast, and keeping the fermentor in a stable temperature for at least three weeks.
Good luck!
Joe