Kana: the basics of writing

Now for the fun part.

Textbooks aren't the way to learn kana, the Japanese alphabets.¹  It's way easier than that.  I'm selling back my Kana from Zero! unused.

Here's what you do.

Hiragana

Making a mnemonic chart

1. Get or make a chart of the hiragana (gotta start somewhere).
2. Get or make a chart with mnemonics to help you remember the shapes.  For example, い, i (pronounced ee), could be a couple of eels:² 
3. Practice writing these characters while using the mnemonics.

There are two reasons I think you should make your own mnemonic chart rather than just cribbing:

1. It'll be full of things that makes sense to you, so that you can remember
2. There are a lot of different arrangements people put the letters in.  I prefer the traditional (shown above).  You can get this by printing the image below and marking it up:
   
   
   

But even so, it saves time to see what others came up with.  Let me Google that for you -- there are tons.

Playing a game

It helps to just make copies of the chart in your spare time, but as you progress, I think you may find this game to be helpful:  the Hiragana Drag-n-Drop Game.  Get this and I think your time learning kana will be cut in half.

Katakana

Now do the same things to learn katakana, all the way down to the Katakana Drag-n-Drop.

Two special problems

In katakana, you'll find some characters that look way too much alike:

シ (shi) and ツ (tsu).  It seems the only difference is in the angles.

I remember it thus:  a burashi (brush) is for brushing up, so if you imagine it to be a brush, the brush itself is written with an upward stroke, and the bristles are kind of flat -- not down.

The second character ツ can be imagined as two surfers diving down into a tsunami.  OK, that's a little scary, but I can't think of any other English words starting with "tsu."

Similarly, ソ (so) is someone sewing down with a needle (left stroke) into a cloth (right stroke); and ン (n) is a nice noseless smile of someone saying "mmm" or "nnn" (image to the right).

Mnemonics for directions of things help for other letters too.  How do I remember if サ (sa) has the curvy thing going back left, or forward?  It looks like a saddle to me.  So I think, "Back in the saddle again."

Now go have some fun learning a secret code that nobody but you can read.  Oh, and about 130 million Japanese.

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¹Effectively, Japanese has two alphabets.  That's not as bad as it sounds:  we have two as well, upper and lower case.  (Technically they're "syllabaries," but let's not get technical.  Thanks to the commenter who pointed this out.)

One is hiragana:  a beautiful, flowing, smooth set of glyphs used for native Japanese words -- おはようございます; どうもありがとう.  

The second is katakana:  pointy, angular shapes used for foreign import words:  オレンジ ("orenji," or orange); インターネット("intaanetto," or Internet).
²Tofugu has a lot more of these images -- check 'em out.