What You Need

Basically, hit up your local brew shop and find a decent kit that contains all the ingredient basics: Malt Extract, Specialty Grains, Hops, Yeast, and usually Irish Moss (helps to clarify the beer). A decent kit explains the basics step by step that you can follow and the beer should turn out OK as long as you do not cut corners. I may be contacted with questions using the contact form, but this site will also have suggestion and links for finding kits and homebrew shops.

Equipment:
1) Brew Kettle. A large pot is necessary for boiling wort. I do recommend an 8 gallon pot or larger. Wort, when beginning to boil, will foam up and so the pot cannot be filled to the top. Allowances for evaporation will need to be made in order to end up with 5 or so gallons of beer. Basic kits are all designed for 5 gallons. 5.5 gallons of water to start will provide for around the proper volume when the boil is over. Some instructions will have smaller 2 or 3 gallon boils, with cold water added at the end. I do not recommend this as if you become more serious about brewing, as this effects hop utilization as wort concentration at time of the boil will be much higher, requiring more hops. I have an 8 gallon, restaurant grade pot made of stainless steel, but we started with a much cheaper ceramic coated pot. It did not last last long, but at least if someone only want to try brewing, it can help quite a bit with costs.
Brew Kettle

Brew Kettle

This 316SS 8 gallon restaurant grate pot was purchased at a restaurant supply shop. Other pots/kettles can be puchased at a local brew shop.
2) Fermentation Vessel. A basic kit usually will include a bucket (with an air lock) that holds more than 6 gallons when filled to the top. This is fine as a begginer vessel, but keep in mind after 15 or 20 batches/uses it usually ends up with some sort of permanent infection that settles into the plastic and cannot be removed thereby requiring the purchase of a new one, and resulting in at least one spoiled batch. Glass carboys are better as they do not have an expiration date, but they cost quite a bit. We had a 5 gallon carboy that was lying around that we used as a secondary at the time along with the primary bucket, but now we use 6.5 gallon carboys for fermenting. Glass is easy to keep clean as it can be filled with chlorinated water and left until needed.  For the beginner making the first batch a plastic bucket is all well and good. I have four 6.5 gallon primaries so that I can have multiple brews going together. The best part about glass though, is that the fermentation may be watched. The yeast swirling around can be seen as well as it can be determined as to when the beer has settled and it is time to bottle.

Carboy with Beer

Carboy with beer

This is a 6.5 gallon primary fermentor, glass carboy. These are best purchased through a shop as shipping will add significantly to cost. A plastic food grade bucket with a lid and hole for an air lock will be sufficiant for the first batch.
3) A Secondary Vessel. A bucket can be used for this that is the same as the primary fermentor, with an air lock, but the same rule applies as above, it will expire with infection at some point. I mentioned above the use of a glass carboy, and a 5 gallon one may be in the budget as a secondary fermentor. I highly recommend transfering the beer from the primary to the secondary after the air lock bubbles only once every 2 minutes or so. If glass is being utilized, activity can be seen and monitored as the yeast swirls around, and also the foam on top can be viewed and will reduce as fermentation winds down. Transerring to secondary gets the beer off of dead yeast (never let a brew sit on dead yeast-make sure a transfer done no matter what after 2 weeks). Again, glass is best, but as a starter, it can be added later and a bucket is fine at the beginning.

4) Wort Chiller. This can be a water bath, but if a big stainless steel pot is being used like I have, it may not be practical. Back when we started, a copper coil wort chiller was only around $30. Now they are over $100. This is a decision that is all about what a person is willing to spend. 316SS chillers are now available as a less cost option. Keep in mind there is something about copper being in contact with the beer at some point in the process, as the yeast needs the zinc present in copper as it is a benificial nutrient for yeast. Copper as a mineral is also beneficial as it will reduce sulfur compounds in the aroma as it binds with hydrogen sulfide (mainly present in lager yeast fermentation). All of the big time brewers use copper kettles. If a beginner wants to take brewing to the next level I do recommend the expense of a copper coil. There are also more complicated wort chillers available that require pumping the beer through the chiller, but as a beginner keep it simple. Hoses and clamps running from the fawcett to the chiller and from the chiller to a drain are necessary as well. Wort needs to be chilled to 70F or so for ales, 60F or so for lagers.
Copper Wort Chiller

Copper Wort Chiller

With the price of copper these days, copper wort chillers have been put out of reach for many brewers. There should be second hand ones around from people that quit brewing or moved on to advanced techniques. A 316SS alternative is also a possablity but copper has properties benefical to beer.
5) Siphon. As a beginner, a plasic siphon is an inexpensive tool for transferring chilled wort to the fermentation vessel. This is something I would recommend looking into as an immediate upgrade from dumping your wort into the fermentor. Otherwise, the dumping method is OK, or if you have a carboy, then a funnel is required.

6) Thermometer. A Floating thermometer is necessary for temperature monitoring during heating and cooling. These are readily available at the brew shop.

7) Bottling Bucket. Unless a brewer wants keg the beer and learn all the ins and outs of operating a draught system, a bucket will be needed along with a siphon hose and canes to transfer beer to bottles. As the fermentors above, plastic needs to be replaced. 20 batches = time for new bottling bucket if care is taken.

8) Bottle washer. Available at home brew shops. required to rinse bottles and carboys.


 
Bottle Washer

Bottle Washer

A bottle washer can be affixed to the sink fawcett and is very helpful in rinsing sanitized bottles or used bottles with sediment. They are readily available at hombrew shops, but make sure that the sink fawcett is nothing fancy and will be adaptable to this useful contraption.
9) Bottles and caps (with a capper). Swing cap style bottles can help here but basic, non-threaded bottles of whatever size are fine. Caps and a capper are available at the shop. This may require much drinking (to make empty bottles!) When choosing this approach, remember to rinse the bottles well as things tend to grow in the bottom otherwise. Bottles can be purchased, but if many batches are planned it may be more beneficial to save bottles after consumption.

10) Bottler. I recommend a spring loaded simple bottler that is fitted on the end of a cane. Ask your home brew supplier.


Bottling Cane

Bottling Cane

This spring loaded bottling cane is what I am using to fill bottles with currently. Simple and recommended.
11) Heat source. This can be a kitchen stove or a separate propane burner that can purchased at a homebrew supply shop (or other source that has outdoor propane burners). Electric stoves are not the best for boiling as the intense heat of the heating element with caramelize and darken wort, which is not something desired in ligher or more delicate beers. In extreme cases, a burnt character can be imparted on beer.

12) A method is required for measuring the original gravity, A kit should come with a hydrometer, or a better idea would be to spend the money on a refractometer.

13) Sanitizer. Chlorine solution works well but rinsing is required. Iodophor does not require rinsing, but should not be used on plastic fermentors. Chlorine will also remove residue after fermentation. There are also other powdered sanitizers available at the home brew shop.
Racking Cane and transfer siphon set up with bottling bucket

Racking Cane and transfer siphon set up with bottling bucket

I prefer a bucket without a hole in the bottom as it is easier to get the beer out of the bottom. Left is said bucket with the racking cane siphon set up. This is not for use with transferring wort, which is a self starting siphon. The racking cane can either be steel or plastic. And that is chlorinated water in the bucket.