Hi, I was just wondering if I were to buy a new iPhone would I be able to use the computers at the apple store to backup my old iPhone and restore onto the new phone that I have just bought?

If you have access to a wifi network, you can use iCloud backup in settings to quickly backup your device before you go in whenever you can. Go to settings, in iOS 11 at the top you'll see your name listed with your Apple ID, iCloud, iTunes, etc click that option. About half way down you'll see iCloud go ahead and click on that. You'll be directed to the iCloud page and after the storage available on your iCloud account you see several iOS apps and at the bottom of that first list is iCloud backup. Select that, toggle on if needed and push "Back Up Now". If you've recently backed up already it'll show that below the Back Up Now button and you may not need to. Then at the Apple store or wherever you are when you set up your new iPhone you will be able to restore from that backup on iCloud. Some of the back up can be restored over cellular data. But a lot of it will need a wifi network to finish backing up.

Ask at the store if they can transfer your files for you.

I don't believe they would (or should) allow you to create an iTunes backup, but they likely will let you use their wifi network to create an iCloud Backup.