You should install iTunes in your new computer (here: http://www.apple.com/itunes/download/) and plug in your iPod. iTunes will then ask if you want to sync your iPod to your new computer. About trasnferring music from your iPod to your new computer, iPod and iTunes does not provide that funcionality. There are however ways to do it. You can read one of them here: http://www.ilounge.com/index.php/articles/comments/copying-music-from-ipod-to-computer
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When you plug in the iPod to the new computer, iTunes should display a message that reads something like "purchased items have been detected on this iPod. Would you like to transfer these purchases?" this message may display before or after you choose to sync the iPod with that computer. However, it may be risky to sync if the message does not display beforehand. It should display a one point though, and a problem only arises if the message is never shown. Hope this helps and sorry i couldn't provide an absolute answer.
You need to have iTunes downloaded on that computer first. Then you just plug it in and it syncs and all the songs and apps are then copied onto that computer.