The process of brewing beer from malt extract is not terribly difficult and requires rather little in the way of equipment besides what one would normally have in a kitchen.
Of course, it's possible to accumulate a great variety of beer specific equipment and use that for your brewing. I'm not going to describe the cheapest possible way to get started making beer. I'm going to tell you how I make beer. Much of what I use for my beer making is dual use, but many things have been specially purchased or constructed for my brewing use
When I am making a batch from extract I start with a kit from Northern Brewer. They have some very nice recipes and put all the ingredients together for you in a box. The ingredients they put in their kits and always fresh and high quality. Some of the other online homebrew stores make similar kits, but I've never tried any of them. I'm happy with the ones Northern Brewer makes and they're pretty close to me so shipping is cheap and fast.
About a day or two before I'm going to brew I pop the yeast packet that comes with the kit. This mixes some nutrient solution in with the yeast and gets it ready to ferment the beer. In about 24 hours the packet will be all puffed up and then I know that the yeast is alive and ready to use. Some people would say that you need to make a starter to build the yeast up to prevent problems. I've never bothered, something I'll have to try one day I guess.
On the day I'm going to brew, I set aside about 3 hours for the brewing process and cleanup. I start by filling my 20 quart stock pot about 80% full with water. I put this on the stove and turn it on high. Then I put the crushed specialty grains form the kit into the provided grain bag. I tie the open end of the grain bag in a knot and loop the loose end over the handle of the pot so that the grain floats in the water.
While the water in the pot heats, I put any bottles of malt syrup that came in the kit into the sink and fill the sink with hot water. Heating up the extract syrup makes it a lot easier to pour out into the pot later.
When the water in the pot steams and the very bottom is starting to collect a few bubbles I remove the grain bag from the pot and throw it away. Then I wait a little while longer until the water is boiling.
When the water in the pot is boiling, I turn the heat off and stir in the malt extracts. I find it helps to put some hot water in the extract syrup bottles to dissolve the part that sticks to the sides. This step requires a lot of stirring to prevent the malt extract from burning to the bottom of the pot. Burnt malt extract tastes and smells terrible and is very difficult to scrub off the bottom of the pot. Then I turn the heat back on wait for the pot to return to a boil.
This stage is when a boil over is most likely to happen. Usually, I stay in the kitchen with a spray bottle in my hand to fight back any foam that forms on top of the boiling wort. I let the wort boil for a while until the foam isn't quite as thick anymore before going on.
Once the foam has fallen back into the mix it's time to add the hops. Usually, the first addition of hops should be boiled for 60 minutes, but it depends on the recipe. When adding hops some foam will form so I usually turn the heat off while I add the hops. Then I set my kitchen timer for 60 minutes.
Depending on the recipe, there may be some other hops and things that need to be added to the boil, so I keep an eye on the timer, but mostly at this point I can relax for a bit. When the timer reads 30 minutes, I put a teaspoon of Irish moss into a small dish of cold water.
After I get the Irish moss rehydrating, I fill my primary fermenter with One Step solution. I put the lid to the bucket, a colander, the airlock and my big brewing spoon into the solution so that they stay clean.
When the timer reads 15 minutes, I put the rehydrated Irish moss into the boil. At this time, I also put my wort chiller into the kettle to make sure it is sanitized.
During the last two minutes of the boil I set the lid of the kettle on loosely so that it gets a chance to be steam cleaned. I also get any hops that need to be added at the end of the boil ready. Then the timer goes off. I turn off the stove and connect my immersion chiller to the faucet and start the cold water flowing. Then I stir the wort into a bit of a whirlpool to get it moving against the wort chiller, put the lid back on and wait about 1/2 hour.
Once the wort is cooled to room temperature, I remove the items from the fermenter and place them on the upside down lid on the counter. This keeps everything that will touch the cooled beer sanitary and leaves the airlock filled with One Step solution.
Then I place the colander over the top of the bucket and pour the cooled wort through it. I do my best to keep as much of the hop residue and other crud from being poured into the fermenter. Then I move the colander out of the way and fill up the fermenter to the five gallon mark with cold tap water. I stir the cooled wort with my long handled spoon to work as much oxygen into the wort as possible. Then I tear open the yeast packet and pour the contents into the fermenter, put the lid on the bucket and insert the airlock. After this I put the bucket in the corner in my kitchen for about a week.
After a week or so, it's time for the secondary. I place my primary fermenter on the counter then assemble the rest of the things I need. For this process I'll need a carboy, a rubber stopper, an airlock and a length of 5/16" hose. I fill the carboy with One Step solution then scrub it out with a brush. I pour some of the One Step solution into a pan and put the other items in the pan to sanitize while I empty out the carboy. After that, I attach the 5/16" hose to the spigot on my primary fermenter and place the carboy on the floor underneath. Then it's just a matter of removing the lid from the bucket, opening the spigot and waiting a few minutes for the beer to drain from the bucket into the carboy. Then I attach the stopper and airlock and move the carboy to a cool dark corner of the basement. Then I clean the crud out of the bucket.
The carboy will remain in the basement for about 10 days until there is no more bubbling from the airlock at all. Once I am sure that fermentation is complete I put the carboy into the fridge for a few days to help the rest of yeast settle out. I find that this helps the beer clear a little bit more before I put it in the keg.
After a few days, I get an empty keg and siphon the beer into it. To do this, I use a raking cane, a length of 3/16" tubing, and a carboy cab. Since I fill my kegs with One Step solution when I empty them, I don't need to do anything to sanitize the keg prior to putting the beer into it. I do make up a small pan of one step solution for the carboy cap, raking cane and tubing. I also drop the lid to the keg into the pan while I'm siphoning. The carboy cap has two short tubes sticking out the top. The raking cane goes through one and I blow into the other to start the siphon. Since I want to keep everything sanitary I always take a shot of vodka first. Also, it's an excuse to take a shot of vodka. Once all the beer is siphoned into the keg, I put the keg in the kegerator and connect the CO2. It takes about a week for the beer to carbonate at serving pressures and I now have about a week left until it has been a month since I brewed that works out pretty well. When the beer is pressurized and in the fridge I clean up the carboy and I'm done with brewing.
Now, comes the fun part. Drinking. I like this part.