Welcome to the forums!
To backup your phone you can use the Kies software on your computer - it'll backup your contacts, pictures/videos (stored on internal memory), SMS/MMS (one of the few reliable ways to backup MMS), call logs and some phone settings.
You should also make sure that your contacts are configured as Google contacts rather than device (or SIM card) contacts.
Be sure you turn off any security so that in case the screen goes completely black you can still access the contents.
BTW, your screen is shot - plan on getting a complete screen assembly replacement or a different phone.