Set up the room.
Having a room to record in may seem so obvious that its not worth mentioning, but it is actually the most commonly overlooked component of a studio. The more space you have to work in, the better. But a big room does not guarantee good acoustics.
For a general purpose recording room, low ceilings, parallel walls made of hard materials, and square-shaped rooms almost always sound bad. High ceilings usually produce a more workable sound.
Additionally, having a balance between reflective and absorbent materials is important. Too much reflective material makes everything sound like a high school gymnasium. Too much absorbent material, and the recordings sound dull and lifeless. In a concrete basement or garage, try hanging up some small carpets. They help soak up some of the excess reverberation.
Also, remember that treating a room to sound good is much different than sound proofing. Hanging up a few carpets will make the room sound better, but will not reduce noise leaking in from the outside world. Finding a space that is relatively quiet from the start is simpler and cheaper than trying to sound proof it later on.
Next, select a recording platform.
This is the device used to capture your audio in some physical form. Some people are still using magnetic tape, DATs, CD-R, stand-alone hard disk recorders and other devices, but the most economical system for most people with a relatively new computer (purchased within the last 2 to 3 years) is to buy software and record to the computer's hard drive.
Computers and software are always changing, but there a few constants. A faster computer is better because it allows you to use more tracks and more elaborate effects. If you have to share your regular personal computer with recording, that is OK, but having a computer exclusively for music is the best option. For many people, a comfortable compromise is to buy an external hard drive that's used only for music. You will need the extra disk space, and the software will run more quickly because it will not have to rummage through all your personal files to get to the music.
Research software carefully. You want to avoid any software that can be used only with a limited amount of hardware. Start with something that works with the computer's existing sound card but can eventually accept an external interface.
Be wary of buzz words in the marketing, such as when software claims to be the "industry standard." That does not always mean it is the best choice for the average consumer who may or may not want to record professionally. Talk to other people with studios and ask what they prefer and why.
Select a microphone.
You can never have too many because they all sound a little different.
If you have no clue where to start, a great first mic is the Shure SM57, because it is inexpensive and it sounds good enough to be used professionally. Once you know what one looks like, you can spot them everywhere!
As an added bonus, they're tough as nails. It's really hard to break one of these things, so they tend to last for many years.
You have a computer and a reliable mic, but you will need to find a preamp to get the mic's signal at an appropriate level for recording.
Many inexpensive external recording interfaces have built-in preamps that work well. This tends to be the easiest option for most people, because it's cheaper and simpler to put the preamps and interface in one unit. However, if you are using a sound card or interface without preamps, you must buy an external preamp that goes between the mic and the sound card or interface. These can be relatively cheap units or very expensive, high-end, "boutique" pieces of equipment.
Get a good set of headphones before you get monitors.
Monitors look great and give that "studio" look, but unless you can afford a set of high quality monitors, and an acoustically tuned room to listen to them in, they will not give you a good representation of what your recordings sound like. Often, recording neophytes will mix for hours, until it sounds great on their monitors at home, only to discover that their music does not sound very good on other sound systems.
Headphones do not alleviate this problem completely, but they do take the sound of the room out of the equation when listening critically. Generally, a balanced headphone mix will sound more consistent from system to system. A good set of headphones will run about the same cost as an SM57 and will continue to be useful as you upgrade your setup.
The last step is to press record and start experimenting!
There are lots of books and tutorials about recording, but its a very hands-on type of activity. Your first few attempts are not going to sound like a professionally recorded album, but do not let this discourage you. Its all part of the learning process.