Seconded Penguin. When I was in Ukraine, I used this book a lot as a beginner.
Once I had a decent grasp on the grammar and some basic vocab, I used Modern Russian. It was a little boring at times, but it taught me a lot of cool rules about the grammar.
The best advice I can give is to try to find a very patient native speaker who understands Russian grammar above the level of an ordinary native speaker. Russian grammar has a lot of nuance and subtlety that is hard to get from a book, and most native speakers, while they have an innate sense of the grammar, will have a hard time explaining it.
The second piece of advice is to get a real grammar book. Skip the beginner stuff that holds your hand. If you understand English grammar, you can understand Russian grammar. Wikipedia is your friend for unfamiliar terms, and you may have to spend some time and effort to really grok some of the more unfamiliar concepts, but it'll be worth the effort. Personally, I used Modern Russian to study the grammar and then carried around a small notebook where I made lists of vocabulary, which I was constantly looking at and adding to. I had lists of body parts, fruits and vegetables, furniture, etc. Anything you could imagine. It was a better ready reference than a dictionary, because it was organized the way I thought. I guess that's piece of advice 2b. :)
Third, don't be afraid to ask questions. I had the advantage of learning Russian in Russia, which it sounds like you may not, but you can still get pretty far with a patient friend or two (or maybe Reddit study buddy). I'm sure I wore out a lot of people with my constant questions, but there are some things that you just won't find in a book. Like how to correctly pronounce ы. I spent a 45 minute трамвай ride with a very patient woman who had me repeat улыбка and лысый at least a thousand times before she was satisfied with my pronunciation. Best education I could have gotten.
That's the best advice I can give. I loved my time in Russia and I still speak it every day. It's been 22 years since I started this journey. I have a degree in Russian, and I still learn things I never knew.
I guess one final piece of advice would be to not give up. You're going to hit some right patches (verbs if motion, anyone?), where you're going to want to throw your grammar book across the room and give up on the whole endeavor. Please don't. Russian is worth it. Не здавайся!
Finally, if you're worried about posting a question publicly, feel free to PM me. I'll do my best to help. Удачи!
Buy this book: http://www.amazon.com/dp/1853993611/ref=cm_sw_r_tw_awdm_x_UpeaybD8PBDGZ (Modern Russian) and start with chapter one. That, plus asking questions of native speakers, is how I learned the grammar.