If you get a decent mic placed properly in the right recording environment, you have overcome most of the challenges in getting good audio.

A decent microphone:Look for one that offers a level frequency response, which means it picks up all sound from the highest to the lowest with about the same sensitivity. Get a mic with an XLR type connection so you can connect to a professional-quality mixer or recording device.

Proper mic placement:This is crucial. The mic needs to be as close to your voice as possible. However, it also needs to be out of the path of your breath or have a pop filter or wind screen, to prevent it from picking up the 'plosives', which are popping sounds made by your breath as they hit the mic diaphragm. Also, the closer your mic is to the sound source, the louder this will be relative to other background sounds.

A good recording environment:'Dead' in audio means as little echo or reverberation as possible, while 'live' means the opposite. Tile bathrooms are the perfect example of a 'live' space; you would usually not want to record in there. The perfect 'dead' recording space would be a wide open outdoor field with 3 feet of soft, spongy moss underfoot. Of course this is not really possible, because places like that have birds chirping and bugs buzzing and moss doesn't grow in the wide open... at least not where I live. So we try to construct small enclosed spaces that soak up as much sound as possible, while suppressing sounds from the outside. I currently use a small bedroom with 2" foam in the window to cut down on the noise from birds and dogs and cars and kids outside. Then I stand mattresses against the walls and hang blankets on any other open spaces. I also take the sliding doors off the closet, because all those hanging clothes soak up a lot of sound. The most important thing to avoid is hard parallel surfaces. If you have two hard bare walls facing each other, or a hard floor and hard ceiling, you're going to get some bad echoes. The point in creating a dead room is so that you can add reverb and other effects later. Also realize that thin blankets will only soak up very high sounds... to soak up boomy midrange and low reverberations you will need something more substantial like sleeping bags, mattresses or 4" thick acoustic foam.

Always monitor your recording with headphones:I highly recommend getting a pair of headphones that cover your ears...not those little earbud dealies. You need this for two reasons: First, you want to block out any outside sounds so that you can monitor just what is being recorded. Second, you want to keep any sounds going to the headphones from being picked up by the mic, such as an accompaniment track you might be singing to.

Now, before you even begin recording put on your headphones and listen to what the mic is picking up. This is when you will hear background noises that you didn't think about before. Especially noticeable will be things like ceiling fans, air vents, outside sounds, the fridge kicking on. I always turn off the air when I'm recording. This can make for some stuffy recording sessions in the summertime in Arizona, but you do what you gotta do. Listening on headphones will also help you to know if you're picking up a lot of pops and breath noises.

People spend waaaay too much time trying to fix bad recordings instead of just learning how to make good ones! An hour spent on setting up the recording environment will save you many unfruitful hours in editing. Before you worry too much about spending a bunch of money on the recording gear and editing software, make sure you have a decent mic placed properly in the right environment!

Recording equipment:There are plenty of good recording setups out there, but I really love my Zoom H4n and use it all the time. It costs somewhere in the $300 range, so it's probably the most expensive piece of a basic recording setup. There are digital recorders that cost much less, if you don't need all the fancy features like 4-channel simultaneous recording. You could also record directly to your PC, and I have done this but there are a few challenges. One is that you don't have a very portable setup. Another is that if you have your computer in the same room as your recording, you're going to pick up noise from the cooling fan. And if you put it in a closet or cover it with a blanket it could overheat. If you put your computer in a separate room, then you've got to run two cables between the rooms: a mic cable going to the computer and a headphone cable coming back from the computer. This is kindof more like a professional studio setup, but if you're a one-man operation then you're going to have to run into the other room to start and stop the recordings, etc. Still, I have done it that way before and survived.

Recording levels: This goes along with recording equipment but that section was so long already that I decided to make a new one. In setting your recording levels you have two boundaries that you must be aware of:
Noise Floor - The noise floor is the background noise present on your recording. All recordings have a noise floor, because it comes not only from the background noise in the room but also from the electronics themselves. There is always a certain amount of noise (sometimes we call it hiss) present in all audio devices. You need to make sure that your input levels are set to be much louder than the noise floor. If you have to turn up the track volume too much in editing, then you will be boosting the noise floor as well and it will become very obvious and distracting.
Headroom - Headroom is the available volume above your standard recording level. When you run out of headroom you encounter clipping or distortion. This is when the device is asked to handle higher levels than it is capable of dealing with.
To properly set recording levels - Get your headphones on and watch your level meters on your recording device. Set your input levels so that you can hear yourself well and so that the level meters are bouncing in a good range. Perform a few bars of your song with the same intensity that you plan on using when you record. Now move to the part of your song that is the very loudest. Sing it in full voice, just as you plan on doing on the recording. If your level meters hit the top or you hear distortion in your headphones, turn down the input levels. Note that it's also possible to overdrive the microphone. This means you can produce a sound so loud that the microphone cannot physically handle it. If you turn down the input levels but the sound still sounds distorted in your headphones you probably need to move a little farther from the mic, at least on those loud parts.

Editing software: There are some good free programs out there. I think one of the best ones to use when you're starting out is Sony Acid Express. This is really just a limited demo of the full Sony Acid program, but it does the basic stuff pretty well as-is. And if you like it, you can upgrade to the full version for less than $100. Acid Express also lets you do MIDI composition and editing, and has some pretty good built-in instrument sounds. I used it to create the accompaniment for 'Michael Come Home' and was pretty happy with the results.

Notes on the recording process:Record everything on separate tracks. If you have a small setup, this might mean you have to record the various tracks one at a time. That's ok! The final mix will be a composite of various takes anyway, and singing harmony with yourself is kinda fun.

Remember that the end product is a result of two things: your abilities and the work you put into it, and you will develop your abilities by doing the work!