Hi Theo,
Please excuse me if any of this is obvious, but just to help clarify things a little, the term you'll run into for extracting CD tracks to music files is "ripping". Also, the various ways for compressing audio are called codecs. MP3 is one of those, not necessarily the best in all situations but it's supported by pretty much any player. The audio compressed using a given codec is then saved in some sort of container file format, which also includes tags that give information about the tracks. The file format for MP3 is pretty consistent, but there are various tagging formats, so you have to watch out for that.
If you're running Windows, I'd suggest dBPowerAMP, http://www.dbpoweramp.com/dmc.htm. It will let you rip tracks with any codec known to man. You'll also be able to check the accuracy your ripped tracks, by automatically comparing your results with a database of other people's rips. This is based on an online database and add-on software they call AccurateRip. It only works if someone else has ripped from the same pressing of the same album, but the database is growing pretty fast, so you'll be able to check the accuracy of pretty much any popular album and a lot of obscure ones. If you consistently get accurate results for CD's in the database then you can be pretty confident about any others.
Apparently they have to charge $14 for a license to rip into MP3, but other than that it's free, and you can use it to convert among any codecs and file formats.
If you have scuffed-up CD's and feel the need to extract every last bit as accurately as possible, you can try the freeware Exact Audio Copy, http://exactaudiocopy.de/. It's a hardcore, cranky program that tends to hang when ripping really messed-up CD's, but for the obsessed it's the best program for recovering accurate data from CD's. It's also the only other ripping program that can use the AccurateRip database.
As for formats, lossless codecs, like FLAC and Apple Lossless, retain all the information from the CD, just packed to something like 65% of the original size. This is fine for storing your files on a big hard drive, but you'll only fit about two compressed albums on a data CD. My CD's are all stored in the attic, and I actually carry around FLAC files on an external hard drive, but that's kind of over the top.
Lossy codecs, like MP3, are designed to discard detail in a way that you won't miss much, but you lose less when you use higher bitrates. 128 kilobits per second is pretty standard for distributing music in MP3 format. The record companies don't mind it that much, because they know it doesn't sound as good as the CD's.
When you're ripping CD's, the only price you pay for using higher bitrates is storage space. Whatever you're playing on, you won't want to use anything lower that 192kbs. 320kbs MP3 will probably sound as good as the original, and it takes about a third as much space as lossless. Other codecs, like OGG Vorbis, will give you more bang for the bit, but you would have to make sure you're player can deal with it.
Have fun with this,
Drew