Just got an new iPHone 6. How do I transfer my contacts when my syncing can't get past stage 2 of 7!

Well that depends. If your last iOS device used iCloud and you enabled contacts in the settings you really just have to sign in to your iCloud account on the new phone. If that doesn't work you'll want to see if the contacts are possibly stored on an email account.

As far as the syncing issue with iTunes, what exactly happens that stops it from getting past step 2?