I come at this from a different perspective than George and Heorhij, since I'm a native speaker of American English and made no attempt to learn either Greek or Russian until I was about 16.
Assuming your goal is to be able to do research in Greek and Russian, your biggest problem with both of these is going to be learning ecclesiastical vocabulary. Most bilingual dictionaries don't include it and if they do it can often be a bit inaccurate. Otherwise learning the essential grammar and picking up a working vocabulary of 1,000-5,000 words isn't that difficult in either language.
For Greek:
You'll want a book geared towards New Testament Greek, since that is the Greek of the New Testament and more or less so of the Septuagint (Greek Old Testament) and Church liturgical texts. For this you really just need a passive knowledge - with a bit of effort you will be able to plow through the New Testament and eventually have a fairly good understanding of the liturgical services in a Greek Church. The caution to throw out there: entirely ignore anything about pronunciation in such a book - in Greece (and by extension, Greek Churches everywhere) older Greek is pronounced with the modern pronunciation system, not a reconstructed system.
The book I used for this purpose: Essentials of New Testament Greek by Ray Summers. It works, but really was not that great of a textbook. Maybe some of the Holy Cross students/graduates can offer some better help here.
For modern Greek:
I worked my way through Routledge's Colloquial Greek in order to have a lot of the basics down. What was lucky for me is that I then was able to spend two months in Greece and was able to use what I had already learned and pick up a great deal more. The problem with the Colloquial Greek book is that solid grammatical explanations are often lacking, so if you want to go beyond the basics you will need another book - wherein lies a huge problem. There are some books around, but they are old. Hence they use the old orthography and have other things to date them. I think there is a new textbook out by Yale, that Anastasios (our admin here) is using. Otherwise once you have the grammar down, just pick up a book of your interests and start reading with a dictionary and working on learning vocabulary. Of course in the meantime it will really help to find someone with whom you can practice conversing.
Russian:
After how easy Greek pronunciation is and in general how easy it is to remember Greek words, Russian will be a bit of an uphill battle.
A great book for starting out is Penguin's Russian Course. The huge drawback is that you will not at all learn how to pronounce Russian nor be ready to really understand spoken Russian at normal speed. But you will have a good grasp on the needed grammar to communicate and an adequate vocabulary. It isn't hard to find Russians around, so if you make a few friends they'll set you straight on pronunciation and speaking. For increasing your reading ability, a good dictionary like Katzner's and finding some books you are interested in is your best bet.
Your options for learning Church Slavonic are a bit limited in English (or German or French for that matter). Basically you can get a grammar and work from that. But you best bet is really just to master a modern Slavic language and work on Church Slavonic through that route.
Which leads me to another point - you may want to consider a South Slavic language over Russian. Serbian is much easier to learn and speak - not to mention the Former Yugoslavia is a lot more fun... think warm beaches vs. tundra....borsh vs. cevapi. The grammar of Bulgarian or Macedonian is VERY easy relative to the other Slavic languages. Church Slavonic is basically the old South Slavic dialect (I'd call it old Macedonian, but that'd cause a row here...) - so in many ways if your research interests are simply to do some reading in a modern language used by Slavic Orthodox, Serbian or Bulgarian would satisfy this need and leave you a bit closer to being able to handle Church Slavonic. If you are interested in Serbian, I can also give you some recommendations for that - just in the last two years some great resources for Serbian have come out.
