To be honest, war with Russia is pretty much inevitable. Although, without using their nuclear arsenal they have no chance of beating the combined powers of NATO without a very long war.
Russia's best strategy would be to launch simultaneous full-scale surprise attacks on Ukraine, the Baltics and Alaska. It’s basically just an extreme version of what worked with the Crimea annexation a surprise blitz, a quick occupation. It’s risky, but the linchpin would be the assumption that everyone in the Western world would be losing control of their bladders for fear of imminent nuclear apocalypse. This territorial expansion would accomplish several of Putin’s core geostrategic objectives (i.e. buffer zone in Eastern Europe, foothold in North America) and put Russia in a much stronger position to defeat the West the next time around. Putin knows Alaska is strategically vital. The Russian Navy is desperate for warm water ports and the untapped oil fields will be crucial for a sustained war effort. Additionally, an attack on the homeland will distract the Americans from the theater in Eastern Europe.
Putin will likely attack Alaska when the vast majority of Americans are completely off-guard: 5:00 PM on the Fourth of July. The invasion will probably be preceded by a massive cyberattack on American and Canadian power grids. This will temporarily bring them offline and delay a counterattack. Wave after wave of bombers destroy airfields and military bases throughout the state. Thousands of paratroopers capture Anchorage and Juneau Red Dawn-style while special forces units take control of critical infrastructure. Within two days civilians are being rounded up at gunpoint and herded into internment camps. The Russian Navy moves to seize control of shipping lines in the Bering Sea. The initial strike cripples America’s military capabilities in Alaska, which falls to Russian control within a week. Any surviving servicemen and Alaskan citizens not in camps melt into the bush and go into full guerrilla warfare mode. The Russians immediately begin moving in the equipment needed to drill for oil on the North Slope.
Meanwhile, in Europe, Russian bombs fall on Kiev day and night. Within two weeks there will be Russian tanks on the Maidan Square. The Estonian and Latvian governments collapse after less than a week of sustained carpet bombing. Putin has declared martial law in Russia, dissolved the duma and outlawed all political parties except United Russia. With the Russian flag flying over Kiev, Tallinn and Riga, Putin will attempt to broker a ceasefire with NATO, but is completely ignored. Both China and India strongly condemn Russian aggression but maintain neutrality.
NATO will likely try and take back lost territories, eventually Russia will be depleted and forced to resort to the final option: Nuclear attack. Make no mistake, Russia has the most nuclear weapons of any country in the world with 8,484 warheads more than enough to do some massive global damage. The world will fall into mutually assured destruction.