Here is a list of gear for a first home recording setup. There are virtually infinite possibilities when it comes to choosing/adding recording gear. An important principle is to make the most of what you have and to add as necessity dictates.
1. A computer: a computer will allow you to record digitally
2. Recording software - otherwise known as a DAW(digital audio workstation.) A DAW allows you to record music with maximum flexibility for editing and postproduction.
Most popular DAWs: Logic, Pro Tools, Ableton, Garage Band
3. An interface: an interface connects the computer with musical instruments or with microphones
three examples of (relatively) inexpensive interfaces: Focusrite Scarlett, Apogee ONE, NI Complete Audio
4. A Microphone - 3 (relatively) inexpensive options: Shure SM57, Shure SM58, Audio Technica AT2020
5. Microphone Stand
6. A pop filter - a pop filter eliminates annoying pop sounds that can occur when singing or speaking
7. Cables - USB cable for interface to laptop, XLR cable for microphone to interface, 1/4 inch cable for instrument to interface
8. Headphones - so you can hear the music back and so you can avoid “bleed” (when sound is unintentionally picked up when recording.)