Does anyone have any tips for packing large amounts?
Research the airline's baggage limits on their website. You'll usually be given a certain number of bags and a weight limit that are included in your ticket, with a charge for additional bags or additional weight. Get a luggage scale (sold at travel or many hardware stores) and ensure you take full advantage of your weight allowance, then determine whether you are able to pay for more as needed. Consider, as Patricia suggests in comments, sending some items separately by mail.
Can we use boxes instead of luggage if it abides by the limits?
We have a question on that here. It's not super-recommended, and the airline may well ask you to sign a waiver where you say that they are not responsible for any damage, but most airlines will accept it (check the policy of your airline in particular though). If you do this, I'd use the most sturdy double-walled boxes you can find; the cargo hold is a rough places for a box, with items shifting and the wheels of suitcases bashing into it, and it's easy for the box to be damaged in transit. Seal tightly, and consider having it wrapped in plastic wrap, but the box might be opened for a security inspection after you check it.
Should Power supplies (specifically for gaming systems and laptops)go
in checked baggage or carry on?
It doesn't matter. Battery packs (batteries you use for an extra charge on your phone or laptop) should be in your carry on, but power cords and adapters can go wherever you'd like.
Will I need adaptors for outlets?
Yes. Russia uses European-style type C and F electrical sockets and 220V power. Check the labels on your devices and power supplies to see if they are dual voltage (able to accept 220V power as well); many electronic devices are. If your devices are dual voltage, you will just need an adapter. If not (often the case for appliances, hair dryers, curling irons, etc...), you will need a suitably sized transformer to convert the voltage in addition to an adapter, which may be quite heavy and expensive, or you can buy a replacement once you get to Russia.