Yes, it is safe to submit your personal information when shopping on the Internet, because you do the same thing when you write a check in a store. The vast majority of the time you will not have a problem. Sometimes you will, but that is not the norm. It is reasonably safe to trust the people who are selling to you.
I think it has become obvious that there is no privacy on the Internet, so submitting information to any outlet can be risky. However, not only are you taking the risk, but so is every other individual in the world. In the long run, you're simply going to hurt yourself by worrying about security threats like Heartbleed.
Yes, it is safe to submit your personal information online. It is safe because companies go through drastic changes to make sure that they are protected against computer viruses, phishing, and online scamming. Companies hire IT department personal specifically to prevent security breaches. When a security breach occurs, the companies combat the breach and become even more secure afterwards.
I think that the Target breech showed us that paying in a store is equally as unsafe as paying with your information online. I think hackers will become more and more sophisticated as time goes on. One should always monitor their credit and report any discrepancies immediately, but there is no way to prevent hackers in a store or online.