IPhone :: Computer Crashed - Backup And Sync To New?

I have the iphone4. My laptop crashed back in February, so that was the last time I backed up my phone and was synced to a computer. I have no way of accessing anything on this computer. It doesn't turn on or anything, it is basically just a shell now, so I can't release my iphone from it's authorization with that computer and itunes. I am trying to finally sync my phone to a different computer. I went to both Verizon and the Apple store and they told me to go under my itunes preferences and make sure that the iphone does not automatically sync to the computer, so I did that. I plugged my iphone in and thankfully it didn't sync... but then they told me to manually save all my information and apps and pictures and contacts and what not from my iphone to my computer so that I had it incase everything got lost. However, there is no way for me to do this!!! I transfered my itunes purchases so it saved most of my apps, but I have no way of saving my music and photos and contacts. I'm not sure if I am supposed to backup my iphone to this computer? Or will I lose everything? And I know if I sync that I will lose everything since there is nothing saved to this computer. If someone could please help me that would be great!!! I have gone to both Apple and Verizon and read around on the internet trying to figure this out, but no one is giving straight steps on how to do this. I don't know how to save my contacts and photos and everything manually to the computer like they told me in case the sync loses everything. And this has been 5 months like this trying to get help and figure this out, and I'm definitely fed up to say the least. I also haven't even been able to update to ios 5 yet so I don't have iCloud or any other way of backing up everything.

Info:
iPhone 4S