In that case, go with iCloud. Go to Settings > iCloud, and follow the setup prompts. This will take care of your contacts. Then, go to Settings > Storage and Backup, and set those options to your liking. Note that if the total used storage on your phone exceeds your available iCloud storage, you will not be able to fit everything into iCloud. If this is the case, you can backup to your computer using iTunes (available for Mac and PC), or purchase more storage at iCloud if you prefer the cloud method.
Once you get a replacement handset, you will set it up from the backup you made, either on your computer or on the cloud.