Before talking about how to quickly pick-up a new language, it would behoove us to understand how to not undertake Russian language studies. Here's a quick list of DONT'S as they apply to how to wanting to become a Russian speaker faster than usual:
DON'T try to learn too much, too soon. Words and phrases will inevitably get mixed up. When it comes to language, cramming does not work. Instead, start with a few easy but all-purpose action phrases.
DON'T get bogged down trying to perfect your pronunciation. The only thing that matters is that native speakers understand you. You can spend years trying to "perfect" the pronunciation of a single word, whereas you should have spent that time adding new words to your working vocabulary. And besides, your pronunciation will improve as you have more and more contact with Russian speakers.
DON'T put all your eggs in one basket, meaning: Use a variety of courses and materials to find which method works best. This is extremely important because it can literally mean the difference between you becoming a speaker of Russian, or quitting altogether. Unfortunately, some of the "big name" courses do not teach Russian well, and are truly beneficial only to advanced students who already understand the complications of Russian grammar, and have a large working vocabulary.
DON'T get discouraged. Sure, Russian can be confusing at times, but there are clear, underlying principles in how all the bizarre grammar fits together. These rules will unfold clearly over time. Just do your best, and realize that native speakers will understand you regardless.
DON'T go in with unrealistic expectations. You're not going to become a Russian speaker overnight. Realistically, you can become conversational in a few months if you use the right course and study the right way.
Enough of the don'ts. What should you do if you want to truly want to pick-up Russian quick?
Learn to speak it even before you start to read it. The reading takes a little time, though there is a course that teaches people in days instead of weeks. Nevertheless, don't put off learning to speak Russian as you wait for the reading part to sink in. After all, that's not how children learn. They start speaking and the written input comes later.
Ideally, you'll find a course that uses both Cyrillic and English letters to sound out the Russian phrases for you. Finally, once you have the right course, make a commitment to studying five or six days a week. Even if for only 15 minutes a day. Get out a calendar and mark off the days when you were able to put in your 15 minutes of study. It's ok to skip a day or two, but shoot for 5 days a week. It's the long-term commitment that will bring you the results.
Use your time wisely. Listen to audio recordings when you are out and about, in your car, or working out.
Look at flash cards when you have down time at any point in the day. If you are standing in line at the bank, do some review. The use of short periods of time that occur during the day will ensure that even if you don't have time to sit down and study quietly, you at least get some input. By studying for 5 minutes here and 5 minutes there, you can get in your 15 minutes of study time each day, or use these short moments of free time to go beyond your normal study routine.
Hope you found that helpful, good luck in your Russian studies.