Can I move or restore content and contacts to a new iphone if my current iphone is damaged?

Yes, if you've been backing up your phone to iCloud or iTunes on your computer.

See this link to set up a new phone from the backup of your phone --> Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

Did read what I provided? If you've backed up the phone, you can set up a new phone from the backup of your damaged phone. The link (words in blue) tells you how to restore a phone from an iCloud or iTunes backup. But that's only an option if you've backed up your phone. If you didn't back up your phone, most of your data is gone, unless you have iCloud sync on for things like Contacts, Messages, Photos etc. And if you didn't have that, then your data is ALL Gone.

You're very welcome. I hope you got what you needed over to your new phone.