No backup process for iPhone 4

I have been trying to set a backup in iTunes for my iPhone 4.
I have already deleted all of the backups, but when I right-click on my iPhone, there was no option for a backup.
I was wondering if the data would be added automatically if I plug my iPhone?
My mother needs a phone for replacement, but only for call and text.
I have an iPhone 2G for her. Will the data plan be automatically billed if she inserts her sim card in the iPhone?
