In regards to iTunes content, an iPhone can be synced or manually managed 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 that were transferred from a different computer will be erased from your iPhone first. Contacts, calendar events, and Safari bookmarks can be synced with the supported applications on multiple computers.
iTunes includes an option to transfer iTunes content that was purchased/downloaded from the iTunes store from an iPod or iPhone, but this is for iTunes content that was purchased/downloaded from the iTunes store only.
First you need to authorize your new computer with your iTunes account with iTunes if you haven't already done so. With your iPhone connected and without syncing, at the iTunes menu bar go to File and select Transfer Purchases From - the name of your iPhone. If a 3rd party app does not transfer, you can re-download the app with iTunes on your computer and you won't be charged again for a purchased app as long as you use the same iTunes account to re-download the app that was used to purchase the app originally.
Photos transferred from your computer are optimized for viewing on your iPhone as part of the iTunes sync/transfer process - the original resolution of these photos is reduced, which is why Apple doesn't support transferring these photos in the opposite direction. Photos on your computer should be included with your computer's backup along with all other important data.
The iPhone is not designed or intended to be used as a backup storage device.
Did you transfer all photos that were available on your old computer to your iPhone? There are some 3rd party paid utilities available that provide for transferring iTunes content that was not purchased/downloaded from the iTunes store in the opposite direction along with photos that were transferred from your computer to your iPhone, but the original resolution of these photos will be lost.
Before syncing contacts with a supported application on your computer, enter one contact in the supported address book on your computer. Make the contact up if needed, which can be deleted later. This will provide a merge prompt with the first sync for contacts, which you want to select. The same applies before syncing calendar events with a supported calendar application on your computer.
