Before you even start recording, there are a few things that you need to understand.
Know what you are recording VERY well
Things will go much smoother and you will be much less frustrated if you can lay down tracks in a couple of takes rather than 20. This includes both knowing the music and knowing your instrument. Guitarists should know their tones before you sit down to record. The drummer should know how to tune his kit. (I can't emphasize enough how important a well tuned kit is to getting a good recording.) The lyrics should be set, and you probably should play a few live shows before trying to record. All this will make things easier when you don't have to change things around during recording.
Don't expect to sound like recordings you listen to
Those albums have hundreds of hours and tens of thousands of dollars invested into them by engineers and producers with tons of other records under their belt. If you are just starting out, then you won't have the experience, the gear, or the sense to make your band sound like Zepplin. That doesn't mean you can't make a good recording of your own.
Don't underestimate the value of a professional engineer
You didn't state if your goal was just to have fun with your friends or to actually put out a recording that you can sell, but if it's the latter, consider the value of a professional engineer to do the recording. They will have gear and experience working with it that will get you a much better sound faster.
After you understand why you're recording and what you want out of it, you can get to the process of recording.
What you are recording on
For someone starting out, the answer is almost always going to be either a simple digital recorder or a DAW (Digital Audio Workstation), both of which have been suggested above. If you have access to a Mac, then Garage Band is included free, which is a good program to start out on. Garage band sessions are also directly portable to Logic (a fairly popular professionally used DAW), which means you may not have to rerecord if you decided to get your tracks mixed proffesionally in the future. Other DAW options are ProTools (the "industry standard") or Reaper. Reaper in particular has a 30 day trial period, after which it suggests you buy a $50 dollar license. However, the program still works fully after the trial period.
What you are recording with
This is gear, and this is where things can really start to get expensive. You need microphones, cables, and a decent interface at the very least. This is a whole different question, and there are plenty of different articles, forums, and arguments on what gear to buy for what. I suggest a number of Shure SM-57's, as you can use them for everything from vocals to guitar to drums to a hammer if you really need to fix the stage. A couple of LDC (Large Diaphragm Condenser) mics as overheads can really help the drum sound, but only if you have a decent room (more on that later.) Bass (and keys if you have them) can go DI (Direct Input) without any problems.
For more on gear, I suggest checking out gearslutz.com. That's where I learned alot of what I know about it.
Where you are recording
This is the last thing that most people think about, but it's probably the most important. You need a quiet environment with a small amount of reverb. Do some research on acoustics. You may need to treat a room with bass traps or foam padding to dampen it for recording. I've recorded vocals in a wardrobe, just because it worked well as a make-shift vocal booth.
The process of tracking
1) Open up your DAW and start a new project.
2) Set up the project settings. I suggest recording at 44.1k sample rate, 24 bit depth. Latency will depend on how fast you computer and interface can run.
3) Add channels for however tracks you plan to record. The number that you record at once will probably be limited by your interface and microphone selection to begin with. Name these tracks as what they will be. This will help you out later when you go to mix.
4) Set up your gear, and set the target channel so it is receiving input from the microphone you want it to. This is called signal flow and is one of the most important aspects of recording. Make sure you can get sound on that channel.
5) Add a click track, and set it to the tempo of the song. This will make your life INFINITELY easier if you have to track instruments at multiple times, a process known as overdubbing.
6) Start recording. Arm the track you plan to record to, hit record, and start playing. Repeat steps 4-6 until you have everything down you want. Make sure to get the best sounds you can during tracking, because that will make the mixing process go much simpler.
The art of mixing
Mixing is much less of a defined process than recording, and all I can do is give you ideas of how you want to go about it.
Usually, you want to create groups for different classes of instruments, then rout the outputs of channels with those instruments into a Aux buss which is the input for that group channel, then the group channel into the stereo buss. That way all of your signal will look like this:
Instrument track -> Group buss -> Stereo buss
This lets you control instruments as groups, and allows you do things like "bring down the drums in the verse" without having to automate 5-10 drum tracks.
Try to open up space for each instrument. You need to do this both spatially (using volume and pan) and sonically (using equalization). You are painting a picture with depth and breadth. Softer things seem farther away. Louder things seem closer. If you have too many things in the same frequency band, your mix will sound "muddy." If you have too much in the high-mids, it will sound "tinny." This is where you listen to recordings you like, and try to make your recording sound as close to those as possible. Add reverb and delay to make things sound big, compress things to make them more consistent in loudness, and also to get them more in your face. Play with everything. As you get more experience, you will learn what to do for each instrument to make it sound the best.
Just remember to have fun with it. That's what playing music is all about.