What equipment do I need to record myself singing or my band?

You need a microphone  to record yourself singing. You could use the one that came with your computer, but we don't usually recommend that. (The quality is very poor.)

We highly recommend a good microphone, mixer and related cables.

USB microphones are relatively inexpensive, but using a low-latency USB audio interface (like a Focusrite Scarlett or something) with a compatible microphone is better if you are going to overdub (listen to playback as you record a new track).