Wow! that is a huge topic for here! You can start simple and with a small budget for 5 gallon batches or you can sink a lot of money into a larger and more professional set up. Basically , you need a primary fermenter vessel, a secondary fermenter with fermentation air locks and some simple handling and testing equipment. hoses, clamps, floating hydrometer, etc.. To get more information to get started well, I suggest looking for a local homebrewers club and supply company and going to one of my favorite suppliers More Beer online and reading their information and looking at brewing equipment, supplies and recipes/kits, books, etc. A more beer kit well executed by a homebrewer will be a good beer that you will enjoy and be proud of. As you learn, advanced or the traditional methods of all grain brewing will likely attract you! The only thing Jimmy Carter ever did I was happy with as our president was open up the law to legalize homebrewing and jump start my brewing experience in 1973. I still sometimes make a More Beer kit beer for a quick and simple brew day! The art and science of Zymurgy will give you a lifetime of learning experiences and take you down many trails if you desire more information/learning, it is a huge and ever expanding field of research and endeavors by curious people the world over. Zymurgy magazine may interest you to look at sometime. A simple American or British ale is a good first beer to try and make your own. Best wishes!