The easiest way to learn anything is going to depend greatly upon the person studying. For instance, I am a very visual learner. When I need to study or learn new material, I write it down and color code the information. I also do very well with flash cards. For the Periodic Table of elements, flash cards might be very helpful. If you are trying to remember the position of all the elements on the chart, your flash cards could double as a puzzle. Practice putting it together over and over until you have it memorized. Another way to memorize the chemical elements and their abbreviations might be a word trick. Try to find something in or about the chemical name that relates to the abbreviation. For example, the abbreviation for table salt or sodium chloride is Na Cl. Chloride is easy to remember because the abbreviation is actually in first part of the word. Na for sodium might take a little more effort. Perhaps trying to associate it with something easy to remember would be helpful. For instance, I used to say "Nah, no salt for me" to remember that Na was salt or sodium.