Recording music at home has become much more accessible in the last few years. It may have been easy for a long time to simply record a simple idea or audio of yourself singing or playing and instrument, but only recently has multi-track recording become something that can be done with much ease and for reasonable prices.
The first thing you need to get to record music at home is a microphone. If you are recording to your computer, you can buy something very cheap or even use many built in mics that come with computers these days. If you decide to use some other kind of analog or digital recorder like a 4 track, you will need to get a proper microphone. Fortunately, this is not hard or expensive. A great mic for starters is the Nady SP-4C dynamic microphone. It costs under $20, is very sturdy, and feels nice in your hands. It sounds surprisingly good for the price, and would work well as a gig mic as well as something to use to record music or audio at home.
Along with a cheap microphone, the only other piece you need is what to record your software on and potentially edit it with. If you want to go the old school path, find a Tascam Porta-Studio 4-track cassette recorder. You can get them for under $100 on ebay or other places. Otherwise, you will probably be using your computer. If you have a newer Mac, Garageband may have come preloaded on it. Use that. Or else, if you like the idea but don’t have a Mac, there are many different alternatives to Garageband for Windows that will do the job well. One more option if you want to be extremely cheap is to download a copy of Audacity which will run on Macs, PCs, or Linux and is completely free.