Studying abroad in Russia is perhaps the best way to familiarize yourself with the language. It can be difficult to learn a language without total immersion, and this way you will be almost forced to pick up the language. Studyabroad.com (see References 1) provides information on studying abroad in Russia and other Russian-speaking countries as well as on Russian immersion programs here in the United States. Most universities in Russia offer Russian courses for foreigners from beginning to advanced, with St. Petersburg and Moscow being the largest destinations.
Taking a Russian course is highly recommended if you can't study abroad in Russia or another Russian-speaking country. A course will keep you on task by meeting regularly and assigning homework as well as having a teacher correct your mistakes. Browse courses at your local university. Most large universities will offer at least basic Russian courses, and some schools, like SUNY Brockport, offer Russian immersion courses from beginnger to advanced. More searching may be required once you have mastered the basics, and a call to a Russian Heritage Center or the nearest Russian consulate may be in order.
A private tutor can be used alone or together with a class to get extra help. Preferably, you'll want to hire a native Russian speaker so he or she can help you get a feel for the culture as well as correct your problems with the language. In addition to highly individualized learning, a Russian tutor will also hold you accountable and help you stay on task.
Study Russian with a friend or find a Russian conversation club through your local Russian Heritage Center, Russian consulate or social websites like meetup.com. This is a great way to get your mind thinking in Russian so you can practice holding conversations in Russian. You can also sometimes find native Russian speakers on sites like Craigslist who are searching for a conversation buddy to trade your English for their Russian.
Although these tools will work if you stick to them, they are probably the least recommended. It is difficult to stick to such a program unless you make up a schedule and stay with it. The advantage to these programs is that they allow you to work at your own pace and take the time you need on things like the all-new alphabet and sounds, which will probably be your greatest challenge. However, these methods are probably best used in conjunction with a course, a tutor or study in Russia.
Rosetta Stone will probably provide you with the most comprehensive independent Russian learning tools (see References 2). It also provides several CDs that will allow you to practice your Russian on the go. Listen and Learn Russian is an introductory course for those on a budget (see Resources 3), and you can browse worldlanguage.com's Russian section for many other resources such as digital translators and children's books (see References 4).
In addition, your teacher may assign you Russian CDs, tapes or computer programs as supplementary assignments if you are enrolled in a class. These can boost your listening comprehension skills dramatically.