Get a chart that compares printed letters to their cursive equivalents. This will show you exactly what each letter looks like when written longhand. If your textbook doesn't provide a chart, look for one online. Alternatively, print each letter and ask a native Russian speaker to write the cursive equivalents beside them.
Copy each letter slowly and carefully. Ideally, use letter writing examples that show the direction of each pen stroke. Choose one or two letters to work on each day. Trace each letter several times until you get a feel for how to write it. Then write each letter five to 10 times in a row. Practice until you can write each letter legibly.
Write out words. When you've practiced all the letters individually, try writing whole words longhand. If you forget how to write a letter, check with your chart. It's better to check than to practice an incorrect letter form. Also, pay attention to how letters connect with each other. In Russian handwriting, certain letters require an extra pen stroke to separate them from other letters.
Get some clear textbook examples of handwriting in Russian. A good Russian textbook should have a least one page of very clear handwriting to give you an easy start at learning to read cursive. Work with this type of writing until you can read it without checking your chart of letters.
Find some samples of authentic, natural handwriting. If you have a Russian speaking friend, ask him to write out a few paragraphs. A pen pal or online language partner can mail or fax you a sample of her handwriting, too. Ask your helper to write naturally and avoid writing bigger or more clearly than usual. Your goal is to get used to average handwriting, which is often less clear than textbook examples.
Work on your writing fluency. Once you know the cursive versions of all the letters without having to look at your reference chart, work on writing full sentences. First, copy printed material in cursive. This will give you a more authentic selection of words instead of limiting you to words you know. When you know enough Russian to write your own thoughts fluently, do free writing practice in cursive instead of print.