iPod Transferring songs to a new iPod

[iMatt mini]

I'm sure this has been asked before, so feel free to bash me with the links, I don't have time to look.

Let's say I get a Nano for Christmas, I already have an iPod Mini with songs on it, would it be possible to put my Nano into my computer and put all my songs on there without a problem? Even iTMS songs?
I ask this because the other day I put my iPod into my sister's PB and tried to get a few songs and they didn't work.

Thanks