Purchase a book with examples of calligraphy script. You can use one of the many traditional calligraphy scripts or create your own original script. You may use a combination of the two. Before you attempt to render a calligraphic alphabet of any kind, use a scrap piece of paper and a ballpoint pen to practice the letters. All you want to do at this point is get a feel for how your hand will move as you form each letter.
Use a calligraphy pen to write the first letter of the alphabet. Do both lower and uppercase letters. Press the tip of the pen gently against the paper, starting at a point most comfortable for you. Try to draw the letter with as little lifting of your pen as possible. The longer your pen keeps contact with the paper, the smoother your lines will be, so only lift the pen when necessary. You want to maintain a fluid appearance in your script lettering.
Use your wrist to change line thickness in your letters. The tip of the pen angled against the paper produces a thin line. You can achieve a thicker line by turning your wrist slightly, which angles the tip of the pen sideways on the paper to produce a thicker line. It will take a little practice to get this technique down, but once you've mastered it, you will be able to vary the line thickness of your script.
Work with each letter of the alphabet. Once you've accustomed yourself to doing the letters fluidly, try forming simple words. Work on writing your name. Practicing often will help you perfect your art. For added flair, try using a quill pen and a bottle of India ink.