How to put music on iPhone with new iTunes?

I have about 25 albums on my iPhone 4, I have my iPhone synced with my old broke-down desktop, I'm trying to put music on my laptop now and it will not let me unless I erase everything and sync with THAT certain library on my laptop. I'm scared, I don't want to delete something I can't get back, what will this do to my 25 albums and library? And just a little curious, can I get album art for an album I bought when I did not have the Internet to find the album cover?

Drag the songs to the upper right where you see the iPod/iphone icon and it opens up and expands

If the music was purchased from iTunes, you just download the purchases to your new computer.

First put iTunes onto the new computer.
Sign in and authorize the computer on your iTunes Apple ID.
Then sign in to the store and go to purchases.
Click on the cloud icon to download all previous purchases to the new computer.

Once iTunes is set up with your music on the computer you can sync to the new computer.

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