As a few people have said above, restoring from iCloud is far and away the simplest way of doing things. No mucking about with cables or computers is required.
I did exactly this with a broken iPad a few months ago (the fact that it's an iPad rather than an iPhone is neither here nor there).
When Apple gave me a new iPad to take home, I fired it up, put in my Apple Id, and asked it to restore from iCloud. It took a few hours to complete, but when it was done my new iPad was more or less a clone of the previous one before it broke.