Somethingofthenight - assuming you have an ordinary windows pc with a cd/dvd drive, yes, you can put any CD in the drive and windows medial player will copy it to your computer. It can also convert files to mp3, which most music players use.
There are lots of inexpensive mp3 players, some look like ipods but are very cheap and have no brand name, there are those made by other electronics firms such as philips, sandisk and sony which are all good, or there are ipods. If you choose to go with ipods, you will need to install itunes on your computer to transfer music files from the computer.
Most players come with earbuds, which are generally useless for kids. If your children are small, JVC make good kids headphones which limit the sound level, which is handy.
If you want speakers/docks instead of headphones, ipods need ipod compatible ones, most other brands use USB connectors. The details for the speaker/dock will tell you if they are compatible with one or both.
FWIW, my kids both had used ipod nanos initially, mainly so they could be used with bedside alarm clock speaker docks, and they listened to audiobooks on them at bedtime. These lasted will - DS's still works and is easily 7 years old now. It just doesn't get used any more as he has a phone now.
Sorry, that was really long, but I was assuming you were coming into this cold!