Here's a good way to remember the first 15 elements on the periodic table. Feel free to sing along. (to the tune of Ten Little Indians)

Hydrogen, Helium, Lithium Beryllium
Boron, Carbon, Nitrogen and Oxygen
Fluorine, Neon, Sodium, Magnesium
Aluminum, Silicon, Phosphorus

And now the symbols. Second verse, same as the first.

H, He, Li, Be
B, C, N, O, F, Ne
Na, Mg, Al, Si,
P!

Students are required to learn fifteen elements with their name, symbol and atomic number. A student may learn any fifteen, but the first are the easiest (and most useful later.) Extra credit may be earned from memorizing more than the required amount. There are many good periodic tables online, but it might be easier to work with a printed copy.

Web Elements clickable table
Pictoral Periodic Table
Periodic table with atomic structure