Home recording systems are a lot like everything else out there in the world today.
The more you spend, the better the equipment
So the things you will probably want to focus on
1) Computer - Need lots of fast storage, preferably 2 HD's. Mac stuff is nice, but you can get more horse power for the $$$ with a windows based computer. Computer will need the ability to take music in by way of audio interface. USB is ok, Firewire worked well but is near end of life IMHO, and Thunderbolt which is more $$$
2) Audio Interface - Tons of options, but $50 to $100 will get you there. If you want to record singing and an instrument at the same time, you'll need to get at least a 2 channel interface
3) Microphones - As indicated a used SM58 will go along way - $75
4) DAW - Digital Audio Workstation. Many different packages out there. As suggested, Reaper for $60 is fantastic. There is a learning curve with reaper though
5) Monitors - Don't rely on cheap headphones. Invest a little money here. Sony and AKG make some good headphones for @$100. Studio monitor are preferred, but can get pricey.
6) Plug in's - You'll want to get some reverb, compressors, limiters, ect. This can cost alot of $$$ - Go to KVR audio free plugins Invalid Link Removed Lots of good stuff there
So, for @$300 plus computer, you'll have a good start.
Problem with computers right out of a store is that they are so filled with bloat ware, your performance is degraded immediately.
If you understand computers, I suggest building your own as you will get WAYYYYY more computer for your dollar