Is there a logical structure to Japanese kana?

I've just begun to teach myself Japanese, and right now I'm trying to focus on learning the kana, currently just hiragana.

Is there any logical structure to how the characters are constructed so that I might be able to identify common vowel or consonant groups which will help me to identify and write them? I'm working on associating each kana with some form of imagery or mnemonic/memory device, but this is easier for some and harder for others.

I'm also trying to gain an appreciation of the different strokes and what might influence a subset of kana to share one or more strokes. For example NA and TA both have the 't' shape symbol in the top left, and NU and ME are almost identical with the exception of the loop in NU. Are the strokes used based on the sound, the meaning, or something else entirely?

Also, if you feel like sharing some basic tips for someone starting out, I'd sure welcome it. I'm currently just watching a few YouTube videos over a few times so that it will sink in, as well as practicing phrases in my head and out loud and writing hiragana on paper and some iPhone apps.

There's no "logical" structure to the kana. There's just as much structure as the English alphabet. The best you can do is memorize them. There are some similarities but these are just a coincidence.