Long, long time ago... I can still remember when...
I learned my first non-Roman writing systems! It started when I was in third grade. That school year was split between Minot, North Dakota and Wichita Falls, Texas. Normally we moved during the summer, but that year we moved during the school year.
I don't remember for sure which school this started at, but I think it was in Wichita Falls. My elementary school library had one book with the Greek alphabet and another with the Russian alphabet. It also had a book with a number of Chinese characters. In all these books, the characters were printed in a very large size, perfect for young readers. I checked these all out repeatedly, and again and again I traced all the letters and characters on reams of notebook paper. I did this because I thought it was fun, but after a while, by the time I started fourth grade, I found that I could actually read the Greek and Russian alphabets! And I could reproduce some (two or three dozen?) of Chinese characters. It was just all that copying that drilled it in, even without any conscious effort on my part to memorize the symbols.
To be sure, I didn't yet know about Greek accents and breathing marks, and I had no idea what the "hard" and "soft" signs in the Russian alphabet meant, but at least I could read the letters that represented phonemes.
My standard procedure now--and the one I recommend to anyone learning a new alphabet--is to copy the whole thing 20 or 30 times in a row. Then start trying to write words and names, anything you like, spelling out the English sounds as best you can in the new alphabet. You will gain a basic control of the characters in a couple of hours this way, at least if it is one of the simpler alphabets like Greek, Russian or Hebrew. Much longer and more difficult alphabets, such as Thai or Devanagari, will take longer, but the basic method is still good. Enjoy!