The periodic table is organized in a way that allows easier memorization than other possible forms. Use visual cues to help your memorization. Try starting with an empty table and filling in as many element names as you can. Over time, you will begin to remember what elements go in what spaces. This strategy helps you learn the positions of certain elements in relation to others.
Some students make the mistake of memorizing all of the elements as a first step. This isn't useful. Your brain can't deal with large quantities of data unless the data is broken up. Thus, to memorize the periodic table with better retention, take it one step at a time. Set a schedule to memorize a certain number of elements in a given time period, say five elements a day.
For memorization of the periodic table to be useful, you need to be able to recall the information. If you have already memorized a portion of the table, it may be a mistake to move on immediately. Frequently review the elements you have memorized.
Your teachers test you on different aspects of the periodic table, so test yourself first. Quiz yourself on the various aspects of the table. Some students make the mistake of memorizing only the elements, without regard for their properties. Confirm that you understand more than just the names of the elements by quizzing yourself on these properties.