The fundamental problem is that Russia does not accept the de-facto status quo where the West is in a position to dictate World affairs. This gives rise to the most tensions where Russian interests are affected. The problems with Russia evolved gradually, it's not true that Putin was intent on obstructing the West right from the start. In fact his initial opinion was that there is no reason why Russia should not join Nato.
To understand how the relations deteriorated, you have to look at the events from the year 2000 till today from the Russian perspective. From this perspective Russia was fighting the war against terrorism in the Caucasus and before that in Afghanistan, only to be opposed and criticized by the West. If we then reject that Russian perspective on the grounds that Russia did violate human rights, there is still the problem that our views about how Afghanistan or an independent Chechnya could be governed and avoid becoming a failed state, was totally unrealistic.
This then causes Russians to see that the Western world view has flaws. Moreover they'll then look at events since the year 2000 from a more critical perspective of the West than we do which then leads them to draw conclusions we typically donlt draw. E.g. the Iraq war was to most of us just an intelligence failure, to Russia this was much more vindication that their attitude toward intervening into the affairs of sovereign countries is indeed correct.
The recognition of the independence of Kosovo was to the West a minor issue, but for Russia this was a grave issue. The West weaseled its way out of this issue by declaring that this was a special case that should not set any precedent. While that was enough to get Western nations to accept this (some countries did not agree with the decision, e.g. Spain, but they were not going to rail against it). But for Russia the recognition of the independence including the declaration that this is not going to set a precedence was totally unacceptable.
How can the West and not Russia have the right to decide on a case by case basis that some region can break away from the country it is part of? Obviously the West would not agree that Russia can do that too.
Then the war in Georgia in 2008 was probably the single most important event that did irreversible damage to the relations between the West and Russia. The Western perspective is that this was an act of aggression perpetrated by Russia. he Russian perspective is the complete opposite. Russia argues that Georgia started the war Georgia and simply lost. While an dependent EU mandated study did fault Russia, one has to be realistic here and accept that the West would in a similar situation not have done things differently that Russia, other than perhaps go for a complete regime change.
One can argue about Russia's peacekeeping mission in South Ossetia and Abkahzia not having been neutral, but then there are plenty of examples of the West having engaged in similar behavior from Lebanon in the early 1980s till Iraq in the 1990s and Bosnia (e.g. illegal weapon deliveries to the Muslims fighting the Serbs).
For the West cry foul play when Russia didn't see itself acting out of bounds of the limits the West itself sticks to, made Russia feel that when push comes to shove, they cannot really do business with the West. Russia did feel that it did accomodate for Western interests. Even when they felt that the West was wrong, that didn't mean that Russia would continuously rail against that.
The straw that broke the camel's back was Russia putting aside all its skepticism about a NATO led mission to police the fighting in Libya (including an arms embargo imposed on both sides) and deciding to not veto a chapter 7 UN resolution to give authorization for such a mission. This ended up with NATO supporting the overthrow of Gaddafi with NATO playing an active role in arms deliveries to the rebels.
And then the events in Libya after that intervention over the last few years were not really positive. We in the West then don't rethink the rationale for intervening in the way we did, the Russians do make that link (no just protecting Benghazi, but the whole enterprise of removing Gaddafi).
You have to consider the complete picture of all these events from the Russian perspective to see how this leads to a de-facto "anti-Western" point of view. It's not that Russians are by definition anti-Western just that they think the Western point of view on many issues is just wrong and the West itself doesn't see that.
This then makes NATO at the borders of Russia much more of a threat. Western influence in countries where there are Russian minorities is seen to be problematic, because Russia doesn't thrust the Western way of thinking to resolve problems. This in turn gives nationalists in Russia the political space to persue their policies. Putin has to accommodate them, but he can't just let the nationalist forces give everything they want.
I don't see any military threat from Russia, not only because Russia cannot project any military power of significance to overpower NATO but also because the conflict between Russia and the West isn't at its core a military standoff. If you go to Eastern Ukraine then the locals there will tell you all you need to know, some are pro-Kiev and some are anti-Kiev and with each day that passes they hate each other more. That Putin is pouring oil on that fire may be true, but leaving out this picture and framing everything on Russian military forces, is giving us in the West a false perspective of this conflict which leads us to see things like a treat to the Baltic Nations that in reality do not exist.
Putin is then making good use of our fears by holding exercises which forces us to spend more on beefing up our military presence in Eastern Europe.