The unfortunate truth about the periodic table of elements, is that there is no fun way to learn it. However, by looking at the basic forms of learning, you can figure out the quickest and most effective way to teach the elements.
The most important part to take out of this is that memorization is the lowest form of learning. Simply having the students stare at the table will not, at all, help them learn the table. No matter how much time they spend doing it, they will never have a deeper understanding of the subject, and thus will immediately forget it as soon as they stop looking at it every day.
A better way to teach the period table is to apply it to other rules of chemistry. The first way you can do this is to teach the students the basis of how the table was created. By going through the basic families and periods of the table, you can better the basic understanding that they have. The families, of course, being set apart by how many valance electrons an element has, and the periods being set by how many electron rings they have.
Another important thing to remember is that by teaching them chemistry, they are going to learn the periodic table subliminally; you really only need to get them to understand how it is ordered so that they can effectively use it as a resource. Then, as you move onto more complex subjects such as the combination of atoms, the students will use the table as their resource. The more that they use it, the more that they will eventually understand the nature of it.
However, if an individual student is having a particularlyhard time learning it, you may need to move to using special tutoring. Seeing as the periodic table is the basis of chemistry, they will need to understand the base in order to build up a tower of knowledge.
One such way to do this would be to give the student 5-10 minutes to look over the periodic table. Then, give them another table with several elements missing. Once they fill them in, give them another period of time to look over the real table. Then, like before, give them a table with more elements missing, INCLUDING the original ones missing. This way they are slowly building onto what they have memorized, and the repetitive writing of the element will improve the longevity of it’s memory.
Once again, I have to stress that memorization should be the last option you choose. It is much better that a student leanrs the table as he or she applies it to situations; it will allow the student to really comprehend what they have lodged in their memory.