Install the Kies software on your PC and (assuming the old phone actually boots normally) you can use it to make a backup of your contacts, SMS/MMS, pictures/videos, call logs and TouchWiz settings.
Then you can connect the new phone and restore that content.
Of course, if you use Google to manage your account, your contacts should already be backed up, and when you setup the new phone it should restore them automatically.
You can use something like Cloud Drive:
http://market.android.com/details?id=com.amazon.clouddrive.photos on the old phone to backup your pictures/videos to the cloud (5GB free) or do similar things with Google Drive or Dropbox.
You can use something like Helium:
http://market.android.com/details?id=com.koushikdutta.backup to backup your apps and app data from the old phone and restore it to the new one.
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