This is really two questions. One is my information securely stored at various sites on the Internet? and Is my information secure as it travels from my PC to a site on the Internet?
The first question about whether the information is stored securely at different sites is a question of trusting the sites. There is really no way for the users to be able to tell how it is stored. You have to trust the company or organization that they are storing your information securely. They should be encrypting all personal identifiable information they are storing. The US government, EU authorities and state governments try to help protect us, but it is reactive protection, after information is stolen that was not securely stored, they fine the companies.
The second question on whether information is securely being transmitted you can determine. Anytime you are entering and transmitting personal information you want the information being sent across the Internet encrypted. By encrypted we mean that there has been an algorithm has been run against the information and it all appears as a jumbled mess. They use a key to do this and the longer the key the harder it is for someone to guess the key and decipher and understand the information that was sent, That is like keys to your house, the more raised and lowered edges, the harder it is to pick it.
When you go to a web site to order something or put personal information in you want to look on tour screen for a lock on it. If the lock is in the locked position then that tells you that when the information is sent, your PC will first encrypt the information, so no one who grabs the signal between you and the server can read it.
In Google Chrome the lock symbol is the address bar at the top of the screen on the left, right beside the box you type in address of pages you want to go to. If it is locked then data will be encrypted.
Another way to tell is look at the first letters on the address of the web site you are connecting to. If it is https:// then it is a secure site and is using the secure HTTP protocol to send information. If it is http:// then it is using the unsecure HTTP protocol and is not encrypting.
If it is just a page you are looking at and not putting personal information in the page like your credit card number, http:// or the lock unlocked are fine. However if you are going to enter information in the page and the lock is not locked or it shows http:// instead of https://, stop, do not do it.