Any MP3 player will let you put on songs from your own CD's. You will either need to use the software that comes with the player to do it, or download a program from the Internet to do this. It's called "ripping"- you rip the songs from your CD's, the program converts them to MP3, and then you use the software/cable/whatever that your MP3 player comes with to transfer it to the player.
A warning, if you have an Apple computer, you pretty much can only use the iPod, nothing else will work. If you have a PC, you have more choices.
My husband has an iPod, I have a Dell DJ. There are pros and cons of both- the iPod is "cooler" and a bit smaller and lighter, but I don't think it's as sturdy as my DJ is. I find the screen on my Dell easier to read than the one on the Mini, but the color screen on the new regular iPod is about even in terms of readability. I like the blue backlight on the DJ, but your kids will probably like that the iPod comes with a couple of games on it.
With one of the smaller, bargain mp3 players, you're not going to get any of those features anyway.
For kids, I might go with the DJ, it feels a bit more solid compared to the iPod and it costs less. The battery on the current models are the same or comparable to an iPod, the older ones last much longer between chargings, but you can't get those anymore, so, don't worry about that much. They have a new player that is like the Shuffle, it looks good although I haven't tried it.
Also, I don't know how good the Apple extended warranty is, but with the Dell Complete Care warranty, they'll replace it or repair it free, pretty much no matter what you do to it, and that includes the headphones and remote control. I know someone from another forum who accidentally dropped their DJ in the toilet and Dell replaced it, no problem. That's pretty good customer service. They've replaced my remote for free after I accidentally walked into a fence and smashed it- ripped the cable right out of it. They had me a replacement within 2 days.
I dropped my DJ from a ladder by accident once and it was just fine after a 6-foot drop to a hardwood floor. Didn't even skip, but one of my friends dropped his iPod on a carpeted floor and it broke.
My husband actually likes my DJ better than he likes his iPod, he likes not having to fuss with iTunes and he isn't tempted to buy a ton of music (on the Dell, you don't have to use the software it comes with to use it on the computer, you can just drag the files from your hard drive to the Dell, so you don't see the ads for new music, etc, that iTunes gives you. Less temptation to spend $$$.) But, he likes some of the iTunes features better than what I have, so, your mileage will vary.
Honestly, I would probably start them with the Dell and see if they took good enough care of it. If they break it or treat it roughly, you'll know they're probably not ready for the responsibility of owning the more expensive and delicate iPod just yet. (This probably doesn't apply to the Shuffle, which can probably hold up to more abuse since it doesn't have a hard drive.)