We live in a somewhat similar situation, though both parents are Russian-speaking with children born in another country. We were advised to make a subconscious geographical boundary while they were young, with only Russian spoken at home (which is easy to do when everybody speaks it) - in particular with the goal that kids can communicate with their grandparents. We were also advised to not worry much about the extra load it puts on the children - and that each extra language added early on can manifest as roughly a one-year developmental delay behind monolingual peers; after the brain speeds up with such extra training (learning to learn, the greatest skill of them all) they will catch up and overtake.
This will be a notable effort on your side, but very rewarding eventually, to help children keep the several languages actively known (and note they are easy to forget if not used regularly - even by adults), and separate (not mixing the two, which is also very hard and millions of people failed to).
Eventually our kids picked up fluent native-level local language in kindergarten and school, and though we did not want to induce too much load - they stepped up themselves to also learn English and then we eagerly helped :)
I'd say they know at least Russian and local languages better than their age-group peers (classmates etc.) only speaking one of those languages, because they had to differentiate and we had the different language rules explained early on. Literally: first-graders here are learning one new letter a week for their first-ever semester, while ours were reading whole books during a lesson to keep not-bored in school. This matched the experience of other migrant families (or mixed-origin parents where still both spoke non-local language at home) who chose to stay with both languages.
For English-speakers analogy, I think properly educated bilinguals do not suffer the modern plague of people genuinely not understanding the difference of "your" vs. "you're" and so on, just because they had to learn the mechanics of languages early on, instead of casually picking something they hear and wondering for a second how that could be written. They would also read more, in different languages, as part of their learning curve - and it would just hammer the right wording structures into the brain. (I as a child had what was dubbed a "natural literacy" in Russian thanks to reading a lot, and had no use for the rules taught in school... until English barged in and lots of same words had suddenly different spellings and after some years confused me; learning Czech much later in fact taught me a lot more about the mechanics of Russian as well, and that was recently reinforced with what our kids learn... so much ado about being a fluent native speaker :) learning never ends).
Unfortunately, we also had acquaintances from earlier migration waves who did not consider knowing an extra language as valuable, and even 12-year old children who came to another country after years of schooling in Russia, can not understand it now as adults and some who wanted - had to re-learn from scratch or close to that. They also lost years of not communicating with their grand-parents, for example, even after the Internet appeared to make that feasible... oops, too late.
There was also a poster-girl in class for how to not do it: her mother came from Moldova (so spoke Romanian and Russian, her parents spoke nearly exclusively Romanian while she mostly speaks Russian to friends) and father is from France. The parents spoke English to each other at home, and she also had Czech from kindergarten. They did not place the lingual boundaries early on, and she brewed a language of her own made from these five - and none of the adults around even knows all five at the same time to fully understand what she emits. They make fairly reasonable guesses, but that is not a good way to communicate all the time :\
Finally, as various famous people said, "you are a person as many times as many languages you know". A well-learned language is not just "can read with a dictionary" but also about knowing another culture, traditions, history, understanding their way of thinking and reasoning. When we had language courses before deciding to migrate, one student was from a Middle-Eastern country and he was close to reaching his goal of learning some 5-7 European languages (roughly, any 3 Romanic and any 3 Slavic and likely some from Hungaric family, to give a sufficient basis to quickly adapt into the remaining languages if needed) to be open and flexible about his choices where to settle and work.
I really hope and wish that you succeed in this endeavour, and notably with the Internet it is a lot easier to achieve than a decade ago - don't forget there are also a lot of online tutors, logopaedics and Russian language and literature teachers, and complete school programs like uchi.ru (not now of course, but in some 3-5 years it will matter), printable materials for setting the handwriting, etc.
