The Russian Empire's expansion towards the east and south meant conquering lots of local tribes along the way (Buryats, Yakuts, etc.). These peoples unwillingly became ethnic minorities in Russia. Which group put up the best fight defending its territory from Russian incursion, in terms of the time or cost taken to subjugate it?
The peoples that put the best fight in resisting Russian invasion weren't those that ended included in the Russian Empire or the Soviet Union, but those that could keep their feet and effectively resisted despite Russia trying a lot.
Some neighbours had wars with Russia but never got conquered in spite of losing some territory, like Persia and China, or Finland in 1940. Others even managed to be powerful enough that Russia didn't actually try to invade them, like Japan - although this could have been different if World War II had lasted for a few weeks more.
However, I'd say that taking in account the Russian effort, the longest and most effective resistance was that of the Turks. The Ottoman Empire and the Turk straits had been a goal of Russia since at least Catherine II, and despite a lot of wars, from the conquest of Crimea to the aftermath of World War I, Russia hasn't managed to this day to reach those goals.
Apparently there is no consensus among scholars on this one, and several Pacific coast cultures substantially resisted Russian encroachment.
Beginning in 1640, the warlike Koriaks of Siberia's northeast allied with the neighboring Chukchi in the most successful attempt of any natives to oppose the Russians. [Lincoln, The Conquest of a Continent, pp. 54]
Stiffest of all was the resistance offered by the Amur River peoples. [Mote, Siberia: Worlds Apart, pp. 43]
... The Kolosh [Tlingits], like the Chukchi of Chukotka, were never fully subjugated by the Russians. [Gibson, Feeding the Russian Fur Trade, pp. 32]
A good question. I would answer that Finland did so most fiercely and determinedly. The Finnish fought the Russians multiple times over the course of history, and would warrant a good example. Finland's presence/occupation during The Great Northern War deserves a honorable mention.
In 1918, a Bolshevik-backed uprising threw Finland into civil war, resulting in a humiliating defeat for the communists by the Whites (the term for Anti-communist military forces during the Russian Civil War. To be fair though, not all of it was a "heroic defense against Russian imperialism"- in the city of Vyborg, 360-420 men were slaughtered, motivated simply because they were Russian (Stalin, in turn, would later commit genocide against the Ingrian Finns himself). Despite that, the Finns would still qualify as a people who fit this question very well.