Need help backing up Iphone

I'm about to purchase a new Iphone, but I havent backed up my old phone in a while. I go to plug my phone in to my computer to sync/back up with itunes and nothing happens. Its like its not registering. However, my phone still charges when i plug it into my computer so i know its know the port or connector.

I take it to my nearest apple repair shop and they think it has something to do with the software on my phone and computer and think they can fix it but i'd have to pay $100 Funds are too low for that right now (I know, even tho im purchaing a fricking $800 phone)

I've never backed my iphone up on the cloud. I'm attempting to do that now but its telling me 4 hrs.. then 5 hrs.. then 6 hrs to complete.

Any other advice?

TL;DR
-iphone wont sync to computer; taking forever to back up on the cloud. Is there another/faster option to recover the stuff on old iphone and put on newly purchased Iphone