Limit the Information You Provide
There are details you'll need to disclose in order to make online purchases, and details you shouldn't have to provide at all. You should expect to be asked for your name, contact information, credit card number, billing address, and shipping address when making a purchase from a legitimate site.
If an online shop requests personal information that isn't necessary for the transaction, such as your social security number or driver's license number, you should be cautious of the site. If you're unsure why a website requests any more personal information, contact them. If everything else fails, it's safest to wander away.