I am Russian citizen and green card holder living in the USA. I am planning to visit Russia for the first time. Do I provide a green card when leaving the US, or I have to show my Russian travel passport as well? And when coming back, what documents they check?
When you check in to fly to Russia, you will need to show your Russian passport to the airline, so they know you can enter Russia. They may want to see your green card at this point as well.
The US does not have immigration officers routinely inspecting the documents of departing travelers, but there can be spot checks. The TSA looks at your identification when you enter the secure area of the airport, to make sure that you belong to the boarding pass you're showing them, but they don't care about your immigration status. You can even use a US driver's license at that point.
When you enter Russia, you will need to show your Russian passport to immigration. They wouldn't normally care about your green card, but if they ask, you can show it to them.
When you check in for the flight to the US, you will need to show your green card. The airline may also want to see your Russian passport at that point; I do not know whether Russian law requires them to check it.
When you pass through Russian exit controls, you will need to show your Russian passport. If they ask about your permission to enter the US, show your green card. No country I've ever left has asked about permission to enter the destination country, but some countries apparently do, at least in some circumstances.
When you arrive in the US, you will need to show your green card to the US immigration officer. They will typically look at your passport, too, since you have it, but as a returning permanent resident, it's not actually required to have it. It's probably simplest just to present them both when you arrive at the desk.