You need some equipment to get started in a home studio, but one super important part of the setup is the room you're in.
Pretty much every room will need some help to sound its best. There's a great tutorial on this over at E-Home Recording Studio. The way you acoustically treat the room can drastically change the way you hear sounds and will affect both recording (if you're doing that in the space) and mixing/mastering.
Theoretically, if you're doing everything "in the box" - you could skip acoustic treatment and just rely on having a really good pair of headphones. But I would recommend against this for a couple reasons:
Your head will get real tired. Headphones, even the best, get tiring to wear, and you'll also tend to turn them up over time to hear your mix. Both these factors mean it can be both more comfortable and healthier for your hearing to mix using a good pair of studio monitors.
Even the best headphones don't really give the same sense of space as a good set of speakers in a real space, aka your studio room.
I achieved pretty good results in my converted garage space with some minimal acoustic foam treatments. Because the garage has stuff (big shelves full of equipment) in it behind my workstation, it serves as a pretty good diffuser.
The next step was to get a carpet pad and through it on the floor. Bare concrete will bounce pretty much every sound back at you so it had to go!
The most important thing most pros recommend, and I'd agree, is the bass traps. These are weird foam triangle things that sit in the corners of your room, and help reduce unwanted reverberations of bass frequencies by "trapping" them. The effect is pretty noticeable. You can hear it if you talk into a corner without a bass trap, then compare that sound to what it sounds like after you place a bass trap in the corner. I used these traps (from Mybecca on Amazon) and they were good quality for the price and work well.
Then I put foam panels in a checkerboard pattern on every large flat surface that wasn't in some way covered.
This is important to reduce the reverb you can get in higher frequencies off of these surfaces. I ordered this set for pretty cheap on Amazon and they work well. The ceiling was the hardest part because it was the largest unbroken flat surface. Because I wanted a non-permanent, non-destructive way to mount them, I just used some small tack nails and double-sided tape. None have fallen down yet, so I'm calling this mounting method a win!
Last, just because my garage space is long and has long corners at the top of the room, I used these corner panels every 3 feet to reduce the reverb even further.
I think by the end I had spent about $200 on room treatment, and it made a HUGE difference. I have done a few recording sessions, and while the room in no way is a professional recording booth, it does well and I can get clean takes through my condenser mic.
But even better than that, when I am doing long mixing sessions, I can hear cleanly what my monitors are putting out and it makes a big difference on the quality of decisions I make while I mix. When setting up a studio room, the modest time and money investment you'll need to do some acoustic treatment will definitely pay big dividends down the road!