More importantly, what was different during this expansion from the times of the Mongol conquest/reign? Something tells me the use of gunpowder by the slower-moving yet more powerful Russians made the difference but I've never been able to find any book or article that described the details of their push east.
You seem to be mixing up a few things here. First of all, Peter the Great only proclaimed his state an Empire in 1721. So it's Muscovy we're talking about here, the Tsardom of Russia (from 1547) or the Grand Duchy of Moscow (before that).
Moreover, the steppes lay to the south of Muscovy, not the east.
The Great Standoff on the Ugra in 1480 is generally considered the end of Mongol slash Tatar rule over Muscovy. At that point the Grand Duchy's main line of defence ran along the Oka river, some 100 km to the south of the capital. Back in 1480, in the run-up to their standoff with the Tatars, the Muscovites burnt down the town of Kashira, in the duchy of Ryazan, as they deemed it indefensible. Three years later, in 1483, Kashira will be Moscow's first gain to the south. Just to give you an idea of how close this was to home, Kashira is 115 km from Moscow and is, well, almost within the city's commuter belt now.
South of the Oka lay the so-called Wild Fields, basically a no man's land stretching all the way to the Black Sea over what is now Southern Russia and most of Ukraine east of the Dnieper. Crimea was a major Tatar Khanate back then (and a vassal of the Ottoman Empire). After they finished off what was left of the Great Horde in 1502, the Wild Fields turned into a major area of conflicts. The Crimean economy basically revolved around mounting massive raids north into Muscovy or the Grand Duchy of Lithuania (the Polish-Lithuanian Commonwealth, from 1569) to capture slaves that could later be sold off at slave markets. Oh, and while they were at it, they would also sack every fortress they could and even managed to burn down Moscow itself once, in 1571 (and almost did it again in 1591).
Muscovy's border stayed at the Oka for a few decades and then they started inching south, building a defensive line of forts and frontier garrisons astride main lines of Crimean advance. These were interconnected with lines of cut-down trees (abatis in the military speak of those days) against Crimean cavalry so the whole affair came to be known as the Great Abatis Line. It was largely finished in 1566 (and did not really stop the Crimeans from burning their way to Moscow a few years later, see above).
In a few more decades the frontier would move south. Then south again. New forts would be built, new lands become sort of safe and available for colonisation. New models for frontier defence would be tested - that's where the Cossacks come in. At some point, well into the 18th century, Empress Elizabeth even experimented with resettling Serbs from the Austrian Empire along the Crimean frontier and giving them some sort of local autonomy in exchange for military service on the border.
All in all, the whole 'push' south took three centuries until Russia (an Empire now) finally conquered and annexed the Crimean Khanate in 1783 and the steppe was won.
Hope it helps. Or were you more interested in Muscovy's drive east across the Volga and into Siberia?
ETA: Sorry, it was Elizabeth who resettled the Serbs in Ukraine, not Catherine the Great. Fixed