The Russian or Cyrillic alphabet, while bearing some resemblance to our own Latin or Roman alphabet, can appear alien to anyone who is unfamiliar with it. If you are interested in learning the Russian language, getting familiar first with the alphabet is a must. One way to do this is by using it to write in your own language.
Become familiar with how Russian letters are formed. The alphabet sample at Freestyle Language shows clearly how this can be achieved. While some letters will be quite familiar and should pose no obstacle, others will look very different and you should practice copying these out until you feel comfortable with them. Note that Russian uses capital and lower case letters in the same way as English. For now, ignore the typewritten letters as these are often very different from their handwritten counterparts.
Listen to the pronunciation. Unlike the Roman alphabet, each Russian letter indicates a specific sound, which is never modified by the letters around it. It is important to become familiar with the sound that each letter symbolizes, and the Language Helpers website provides an excellent facility whereby you can press the relevant symbol to hear it pronounced correctly. Listen to these as often as necessary until you feel confident that you recognize what sounds are represented by the letters.
Write frequently. It takes hours of practice to get moderately proficient, but you can make this fun. If you keep a journal, try getting into the habit of writing it using the Russian alphabet. Or, if you prefer, just find any piece of text, such as this article, and rewrite it. It may be slow in the beginning, but with perseverance, you gain confidence and what once seemed like inaccessible code will become second nature.
Start joining it up. Like most people brought up using the Roman alphabet, you probably spent a long time writing freestanding letters. Your first few attempts at writing in the Russian alphabet should be the same so that you get a good hold on the shapes you are producing. Once you've come this far, you'll discover that, in many instances, the way one letter joins on to the next becomes obvious. Should you need a little more help, the University of Oklahoma provides a useful PDF which can be downloaded from the resources section.