The premise of the original post is already wrong. Western-to-mid Europe did not always distrust Russia, even though xenophobe movements on all sides of the borders try to make it look that way, even going as far back as Alexander Nevskyi and the crusades.
If we start with the early modern period, when Russia reappeared as a major state, France was in bitter rivalry with the English and the Holy Roman empire. The French were thus traditional allies of the Ottoman Empire, which was fighting the HRE, which did not have conflicting interests with the Russians. Which is why the relations between those countries were quite good. To the point that foreigners in still somewhat isolated Muscovy were collectively called "Germans" for a while.
But the HRE didn't fight the Poles much, which the Russians were fighting a very bitter and existential struggle against (Poland-Lithuania were the ones that came closest to annihilating it for good, and were rather genocidal in their warfare - in some ways worse than the tartars/mongols). So there was no alliance there. The English on the other hand became very dependent on Russia, the Muscovy Trading Company, chartered in the mid-16th century, was the first chartered joint stock company, and thereby an important step towards the capitalist future to come. The relationship wasn't always easy, and there were negative things written about each other and the Russians being outraged at the execution of king Charles I., but that alliance held in general until the 18th century, when two developments coincided and caused a complete reversal:
1) England now eclipsed its rivals by a large margin, especially after great gains against the French in the 7-years war, and now had significant interests in the Mediterranean themselves.
2) The Russo–Turkish War of 1787–1792 established Russia as a far superior power to the Ottomans, which were no longer seen as a major threat. It is also the war that saw Russia gain Crimea, which remains a crucial focal point for all geopolitical strategies.
Russia having risen so far and the other traditional great powers having declined significantly, meant that England now saw Russia as the main threat and completely reversed the trend.
They always trusted Russia enough to see them as valuable allies whenever needed (e.g. Napoleon), but when they weren't needed as such, they were seen as the great game to be hunted down.
Halford Mackinder's heartland theory didn't help either.
Megalomania, avarice, the difference in cultures and chauvinism all fueled the Russophobia which remains a common tool for the hawks til this day.
It is no wonder that Rudyard Kipling, the author of the
"white man's burden",
hated to see Russia as part of the enlightened west:
The French meanwhile continued their traditional century-old alliance with the Ottomans, which they had started because of the Holy Roman Empire, and were thus also pitted against the Russians, and after the revolutions, had quite bad relations to Russia because of those as well. France only had really good relations with Russia after they'd been decisively defeated by the Germans in 1871, and only until 1917. The French invested a lot in Russia between those dates.
Speaking of Germany: Prussia would've seen an early end to its aspirations if it hadn't been for Russian aversions to the Habsburg war aims in the 7 years war. It wasn't, contrary to popular belief, the death of the empress that saved Frederick the Great, since the empress herself also had been quite averse of where the war was going.
The Habsburgs meanwhile were always opportunists. Russia saved them from almost extinction in 1848, Austria immediately went to backstab them in the Crimean war a mere 5 years later, "surprising the world with its ungratefulness", to quote a contemporary ("eilte die Welt mit seiner Undankbarkeit zu verblüffen").
The Crimean war was also another important milestone in European history, as it ended the
holy alliance, which specifically adressed the problems and mistakes of nationalism that'd lead all the way to world war I.
To finish the list with a very positive example: The Russo-Danish relations remained very solid throughout the existence of the tsardom, mostly because of their common rivalry against the Swedes.
Sure: The fact that most of the Western countries failed to get a clear, realistic picture of Russia's capacities and capabilities didn't help. And to a certain degree, that hasn't changed til today either. To quote
a very recent RAND study on how to destabilise and mess with Russia with nonviolent means:
But nothing has changed over the years in the regard that the relationships between Russia and the West were far more nuanced than they're purported to have been.
A couple of months ago, I saw an article from a local newspaper from my region, detailing how Russia donated a very significant amount of money to deal with a famine the region struggled with.
The first step to change is usually up to oneself. Educate yourselves, have clear moral principles, don't be chauvinist.
Everyone knows racism/antisemitism/xenophobia isn't ok. Russophobia isn't either. Don't fall for the trope that Russia is basically a huge Zerg hive. It is not.