Learning how to brew can be both overwhelming and overcomplicated. BrewNewbz will keep things simple. Below are two suggested shopping lists for getting started quickly. If you want to just try brewing out once or twice for fun, go with option 1. If you want to pick this up as a hobby, go with option 2.
Purchase a Brooklyn Brew Shop kit and make sure you have a few extra supplies on hand. This is the easiest way to get started on a real all-grain homebrewing experience. It will only run you $40 on Amazon. The kit supplies you with nearly everything you need to get started and comes with easy to follow instructions. This is the go-to option if you're not certain that you'll stick with homebrewing in the long run, if space for brewing equipment is a concern, or if you just want to try brewing as a novelty. There are other kits out there, but I recommend this one for its authenticity.
In addition to picking up the Brooklyn kit, you'll want to make sure you have the following items on hand for brew day:
Stirring Spoon. You'll need a long spoon that's easily sanitized for stirring the grains within your stockpot.
12-Quart stockpot. A 12-quart Stainless Steel Stockpot will hold enough to brew the Brooklyn kit with room for your grains and boiling. You may want to pick up two of these if you don't already have another sanitized container that can hold a bunch of boiling-hot liquid as you remove the grains through a strainer.
Already brewed a simple batch before? Convinced that you're going to stick with homebrewing? If either of these are true, you're probably better off with a more advanced kit. You can pick these 5-gallon kits up on amazon for anywhere from $70 - $500 depending on how fancy you want to get. The picture to the left is the brewing starter kit that I recommend for getting started ($68 with free shipping on Amazon). Of course, you can also source the parts individually if you already have some parts that you see in the kit. It's usually more expensive to buy everything separately though.
Once you have this kit, there's many methods of brewing that you can pursue. I advise the brew-in-a-bag all grain method. This has the authenticity of "real brewing" but cuts costs and time spent per batch by removing the need for a few other peices of equipment. You can then easily scale up from here if you want to buy new brewing equipment, or continue to use this kit if you want to maintain a budget.
If you go with this kit, or one like it, I suggest picking up these additional optional items: