If you get an Apple product you will need to install itunes on your computer in order to put music (etc) onto the player.
Personally, having at last got my hands on a computer that no one is allowed to install itunes onto, I would rather have my fingernails pulled out one by one than have anything to do with Apple and itunes ever again.
OTOH, if you go the mp3 route you can just download music from, say, Amazon into your 'music' folder, connect the mp3 player, open the player as a second window (to 'music') and drag the music onto the player. You can also use Windows Media Player to organise your music if you prefer.
An iPod is technically an MP3 player in that it and the other devices you mention play MP3 files. MP3 files are just the file format of the music file and tell the computer how to interpret the file.
The iPod is Apples version of an MP3 player and is intended as a premium product which is why you'll end up paying slightly more or you could pay less for another MP3 player which doesn't have any extra features but will play MP3s none the less.
There are several different versions of the iPod ranging from an iPod Touch which can play games, email, surf the net etc or the iPod Nano/Shuffle which are more geared just for playing music.
There are several ways of getting music onto your MP3 player. If you buy an iPod you can buy songs through the itunes store and then transfer them onto your iPod. If you have MP3s already stored on your computer you can import them into your itunes library which is fairly painless.
You could also download download MP3s from a online source like amazon or if you have a physical CD you can copy (Rip) them to your PC and then copy them over to your MP3 (including iPods).
If you have a look on itunes and amazon you can see how much individual song/albums cost as they tend to vary.