Music on new computer
Not when just connecting your iPhone to iTunes on the new computer. If Sync Music is not selected under the Music tab for your iPhone sync preferences, no music will be erased from your iPhone. The same under the Video, Podcasts, and Applications tabs for your iPhone sync preferences with iTunes on the new computer.
In regard to iTunes content - all iTunes content including music, video, podcasts, and 3rd party apps, an iPhone can be synced with an iTunes library on a single computer only, and photos can be transferred from a single computer only. When transferring iTunes content and photos from another computer, all iTunes content and photos transferred from a different computer will be erased from your iPhone first.
You can sync contacts, calendar events, and Safari bookmarks with the supported applications on multiple computers.
You need to transfer your iTunes library to your new computer along with all other important data that you will need or want to access on your new computer. You will also need to authorize your new computer with your iTunes account with iTunes.