The first decision that needs to be made when starting a home studio is what interface you will be working on. This means choosing between computer recording or some other medium. Many times this decision will be made for you as you will probably end up using what you have around you. Meaning that if you already have a nice computer to work on, you will probably take this route. In my opinion, computer recording is the way to go as it opens up the door to endless possibilities that would not be possible otherwise. For the most part in this tutorial, I will be assuming that if you are starting a home studio that more times than not you're talking about recording with computers. The only other option to consider would be a portable all in one recording device, like something that Tascam will make, but this limits your options a great deal.
Once you have your medium for recording, no matter what I would suggest getting some sort of recording software to use. Even if you aren't recording with a computer, editing and mixing is just much easier inside a computer and for me there is no other way to do this. However, if you are going to go the Pro Tools route (which I would wholeheartedly suggest), then you should definitely be using a computer for everything because you need an external audio interface anyway to run Pro Tools. Having a Pro Tools rig makes it possible to bring your sessions from larger studios and move them back and forth easily. Most studios will have Pro Tools and for this reason this would be my first suggestion. However, I understand that this is a bit more costly than other rigs and I would suggest shopping around to see what the best audio interface and software combination will be for you.
The most important thing to remember while setting up your home studio is that the best way to get good recordings is to get a good sound from the source. This is extremely important when talking about a new home studio because you probably won't have the best sounding recording equipment to start off with and placing an extra emphasis on your source sounds will take you a long way. Even if you have the best recording gear in the world, it won't matter unless your original sound is good. Of course this means having good instruments and good players, but I'll assume that if you are building your own studio that you probably already have this...
Once you've got your basic audio interface/computer set up picked up, it is time to start thinking about microphones. Now more than ever the market has to offer a wide array of all different types of microphones at extremely cheap prices. It is definitely possible to get some great sounding microphones on any budget. As long as you aren't planning on recording drums right away, you probably only need a few microphones to start off with. Having even just a single dynamic and condenser microphone will bring you a long way for home recording. A few suggestions I would make would be to check out Rode's microphones for a condenser and you definitely can't go wrong with a Shure SM57 for a dynamic. There are tons of options out there and I would suggest doing to research to see what will work best for you and what you want to record.
Lastly, you should also be thinking about different effects and signal processing. This can be obtained either in the form of a plug-in or a piece of outboard gear. While in my opinion the best plug-ins can't touch the best outboard gear, plug-ins can be a cheap and easy way to add some signal processing to your sound. Pretty much all recording software will come will some basic signal processing and this will give you a good start, but as time goes on it is definitely a good idea to branch out and look into some better sounding plug-ins as this can make a big difference in your mixes.