I don't know how easy this would be considered but I would take a two prong approach to this.
Prong 1: Moving your music over could be accomplished by following the steps in the
Apple support article. You could also us any external hard drive if you don't have a spare iPod or use the CD/DVD method that links from that page.
Prong 2: Moving your address and calendar. Now this part I haven't done myself but I don't see why it wouldn't work. When you hook up the iPhone to your new computer and iTunes opens click on the Info tab for the iPhone. From there you check the boxes to sync your calendar and addresses. Just don't check the boxes (towards the bottom) to overwrite the data on your iPhone.
That should do it but keep your original computer handy just in case I'm missing something and you need to restore your last back up. Otherwise once you're done just make sure to de-authorize the old computer if you aren't going to use it anymore.