How to move household stuff from Russia to USA

You usually start with a moving company and a shipper. They deal with the customs people at either end.

Thanks, will search for a moving company

The hardest parts of the logistics involved they will take care of for you.

I've moved internationally twice in my life so far. There is a LOT involved... but the people you hire take care of most of it for you, and will inform you when you need to do anything. This is a case of money well spent.