Some people can commit songs to memory easily. In the first two links below, are periodic table songs that can aid you in memorizing the periodic table of elements. Another method can be seen in the third link from About.com chemistry. It uses mnemonic devices to help memorization become easier. If you remember the phrase, and know the element symbols, you can memorize the order of the elements. For example, He Lies Because Boys Can Not Operate Fireplaces stands for Helium, Lithium, Beryllium, Boron, Carbon, Nitrogen, Oxygen and Fluorine. This represents elements 2 through 9. Try making up some more mnemonic devices of your own to ease memorization. Good luck.
Try making an acronym and memorizing it. As for me i made a bunch of flashcards and studied them, but a song could also help.
As far as elemental names go, its just memorizartion or you could try to make sentences out of each first letter in a row or column. songs work well too, some people like to draw things out so drawing the periodic table for practice would be a good idea as well.