A secure website is one that uses encryption. When you send your credit card information, the numbers are encrypted so that they can't be detected as actual numbers between the sending and receiving computers. At the receiving end, the encryption is decoded again.
You can tell if you are on a secure website by looking in your browser bar (where you type your web page addresses). Look for an "s" on the end of the http (like this -- https//). Then look for a little padlock symbol at the bottom of your web page. If both are present, it is a secure website.
But this will not save you from a keylogger, I'm afraid. As the user types, the keylogger collects each keystroke and saves it as a text file. Later on, the third party person that installed the keylogger can take note of whatever was typed, and use it.
You need to be sure that you have completely got rid of any keyloggers on your system. When you say you deleted them, did you use software specifically designed to get rid of keyloggers from your computer?
Just make sure your computer is now clean and keylogger free, and when you make up a password, use a combination of capital letters and small letters. Also some websites have a drop down menu where you pick numbers or letters for your password with your cursor, so a keylogger can't know what was chosen.