Things haven't changed from Russia's perspective since the breakup of the U.S.S.R. Russia still wants to control all of Eastern Europe. Russia thinks that they have a right to shoot down Ukrainian planes and the like. Trump doesn't pull any punches and he will make sure that Russia knows what it can't do.
Russia respects one thing: strength. Trump may be a buffoon, but he - with the help of his advisors - will present at least the idea of a stronger America. Plus, Biden is incorrect: China represents the largest threat to the international order. Russia has a stake in the current game. They are a poor yet large country who has managed to cultivate a sphere of influence. China, while economically and even militarily stronger, has less influence, and will seek to change the rules of the game.
It seems clear that Russia is in some ways pandering to Trump and had some involvement in his winning the election, so I think that it is unlikely that Trump will confront Russia. I think that it is a losing political strategy to appear to align with dictatorial heads of state, but this would appear to be one of Trump's current strategies.
President Donald Trump will not likely take a hard line against Russia or Putin. Russia is a threat but not our biggest threat. The Islamic State is a greater threat to the international community. Terrorists attacks are more likely to destabilize the world then Russia will. Trump will need to defeat the Islamic State.