"So, I can buy albums off Amazon without a subscription by choosing the mp3 version. I would then need to download that mp3 onto my laptop, and then onto the mp3 player, right? "
Yes.
"Or, I can buy an actual physical CD, burn that onto the laptop and then onto the mp3 player, is that right? Slightly illegal but hey, we all recorded tapes in the 80s. "
Yes, and yes it is technically illegal again.
"What's the point of an mp4 player? I know it's for video rather than just audio, so do people use it to watch films? Or are some songs available as an mp4 to include a music video?"
MP4 is technically a type of file that can be audio or video (or other things). But some players will use "MP4" to mean that they can play video. Places like iTunes do sell videos.
"I know what DS would love to do, is to download his music from youtube but I'm not sure if that's possible."
Yes it's possible, but it's also illegal and much worse than just converting the CD in my opinion. Don't do it.
The Sony MP3 player is an example of a player that won't work with things bought from iTunes and I also think that it will only work with Windows computers. Those things may or may not be a problem for you.