Really curious to see how this all plays out. If Trump really does want to be a full ally of Putin, so much will change. Further Russian intervention in Eastern Europe will be probable. Concerns over an "oppressed Russian minority" could pretty conceivably lead to an occupation of Estonia. The bloodless incorporation of friendly former Soviet republics (namely, Belarus, Kazakhstan, or even Azerbaijan) is also a lot more likely.
On the other hand, aggression against Iran will probably be off-the-table. Putin is far too friendly with their government to let Trump to beat that drum. American imperialism in general will be neutered. While Russian influence in the Middle East and Asia would skyrocket, Russia probably can't afford the kind of political domination America tends to assert over its third-world dependencies.
The role of Europe will also change enormously under a Russo-American alliance. NATO would become both far weaker and far more necessary. Periphery members like the Baltic states or Ukraine might be ejected to avoid conflict with Russia. Both France and Germany would have a renewed military significance, with one of them potentially becoming the new commander-in-chief of western liberal democracies.
