Possibility to recover data from restored iPhone? My iPhone4 said it needed an update to the new softwear so I went to settings and hit update. 
Then it went to a screen that had a little iTunes box and  connection cord underneath it. 

It stayed on that and wouldn't shut off or restart. I read online I needed to download iTunes to get into recovery mode which I did when it was done and my phone finally came back on EVERYTHING was gone.  

Is there anything I can do to get my contacts/ pictures back?
Unless you had a backup, no. Do you have a copy of your contacts anywhere? iCloud for example? Visit www.icloud.com and login with your AppleID, see in the contacts tab if you still can find your contacts.

If this isn't the case, I don't think you can do anything but learn out of it and make backups regulary.

iTunes is offering you this option. See the support document on iOS: How to back up and restore your content for a complete description.