There's an easier way, at least up to element 18 (argon).
If you just stick all the symbols for all the elements together it makes a big word:

HHeLiBeBCNOFNeNaMgAlSiPSClAr
pronounced
H-Heli-Beeb-Canoff-Nenarr-Magal-Sips-Clar.
Well, it works for me.
When I was at school, ooh just a few years back ahem... I had a chemistry test which involved knowing the first 20 of the elements of the periodic table off by heart. Luckily my brother had taught me the following mnemonic.

Once you learn it you can never forget it, well at least I can't

So here it is -

The easier method of memorizing the first twenty elements of the periodic table:

For all the Him & Hers -
Little Betty Bright Child, Nurses Our Friends Neatly. Nasty Magy Always Sings Pop Songs Clearly Around Kirsty Carol

I'm afarid you're just gonna have to remember the first two elements the old fashioned way, or the lame way I've stated above...

It all corresponds to the following -
H : Hydrogen
He : Helium
Li : Lithium
Be : Beryllium
B : Boron
C : Carbon
N : Nitrogen
O : Oxygen
F : Fluorine
Ne : Neon
Na : Sodium
Mg : Magnesium
Al : Aluminium / Aluminum
Si : Silicon
P : Phosphorous
S : Sulphur
Cl : Chlorine
Ar : Argon
K : Potassium
Ca : Calcium
I was always impartial to just memorizing the word:

HHeLiBeBCNOFNeNaMgAlSiPSClArKCaScTiV, which is pronounced:

Huh-He-Lie-Be-Buhc'inoff-knee-Nam-galsips-clark -kickass -cutiv.

Stupid, but effective.

In fact, I have decided to name my first child this, in order to make her scientific knowledge immediately aided by her name.

Or maybe I'll just go ahead and call her:
"The-square-of-the-hypotenuse-is-equal-to-the-sum-of-the-squares-of-the-other-two-sides".

The memorization of nmemonics, in both cases, is rendered completely redundant! What could be easier to remember than your own name?

Some people, I suppose, would say I am stupid, but effective.

In Japanese, they have mnemonics for groups (i.e. columns) on the periodic table as well. As you might expect, they are as weird and womanizing as English scientific mnemonics. Here are a few of note, stolen from a Wikipedia article:

Group 1 (H - Li - Na - K - Rb - Cs - Fr)
Ecchi de ricchi na kanojo wa, rubii o seshimete Furansu e.
The rich perverted girl stole some rubies and went to France.

Group 2 (Be - Mg - Ca - Sr - Ba - Ra)
Beddo ni mogureba kanojo no surippu barairo.
The girl hiding in the bed has a rose-colored slip.

Group 15 (N - P - As - Sb - Bi)
Nippon no asa wa subuta to biiru.
Japanese mornings call for sweet and sour pork and beer.

Group 16 (O - S - Se - Te - Po)
Osu no seiki teppou.
Male genital gun.

Group 17 (F - Cl - Br - I - At)
Fukkura burajaa ai no ato.
Swollen brassiere after love.

Group 18 (He - Ne - Ar - Kr - Xe - Rn)
Hen na neesan aru hi kurutte kisu renpatsu.
My strange sister went crazy one day and started snogging like mad.

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