What social or military innovations allowed the Russians, who were long vassals of the Golden Horde, to eventually free themselves of the "Tartar yoke" or could steppe rule of Russia be characterized as oppressive? At one point did Russia eventually gain the advantage over the Mongols and conquer their successor states? Was this simply a matter of Tartar weakness and disunity or did the Russians possess social or military developments that allowed them to surpass the people of the steppe?
There's two parts to this question, the first being how Muscovy/Russia ended its tributary status to the Golden Horde, and the second how Muscovy/Russia expanded its own control over the steppe.
I'll have to go over the first part a bit briefly, as it's not really an area of Russian history that I'm a deep expert in (and so I don't want to discourage any further answers here). The long and short is that the Grand Duchy of Moscow was a tributary to the Mongol-successor state known as the Golden Horde, which was based at Sarai, a city on the lower Volga. The Muscovite Grand Dukes, like other Russian vassals, were confirmed in their office by the Khan, and collected taxes on the Khan's behalf. The history of Muscovy's relations with the Golden Horde is long and complex, but to summarize, it varied a lot between good relations to warfare. Dmitry Donskoi (ruling in the late 14th century) is traditionally considered to be a big starting point in the waning of Mongol/Tatar influence, notably with his victory at Kulikovo Field in 1380 (although it's worth noting that Moscow was sacked by Khan Tokhtamysh two years later, and Dmitri submitted to his authority, so this is a bit of back-projecting a national narrative). Tribute finally stopped under Ivan III, and the Golden Horde basically was unable to reinforce this status after the 1480 Battle of Ugra River.
Some important things to keep in mind about that century in between is that it saw the fracturing of the Golden Horde through many civil wars into a number of successor states, the most notable by the late 15h/early 16th centuries being the Khanates of Crimea, Kazan and Astrakhan. Kazan and Astrakhan were ultimately conquered by Ivan IV (the Terrible) in the 1550s, and the Crimean Khanate became an Ottoman vassal, taking over the Black Sea slave trade and being in conflict with Muscovy/Russia until its final conquest in the 1780s. During the 14th and 15th centuries, the Golden Horde also saw the Grand Duchy of Lithuania expand into much of its territory and vassalages in the former Kievan Rus', so Muscovy alone wasn't doing all the expansion at the Horde's expense.
OK, so that's a brief timeline, but now on to Russian expansion into the steppe proper.