Short answers, pulled from the story.
Between 1500 and 1800, Russia brought the steppe and forest-steppe of Ukraine and southern Russia under its control. Key acquisitions included the Khanate of Kazan (1552), Astrakhan (1556), Left Bank Ukraine by the Treaty of Andrusovo (1667), Zaporizhia by the Eternal Peace Treaty of 1686, the port of Kherson (1778), and Crimea (1783).
The Belgorod Line was a continuous fortified barrier built beginning in 1646 to protect Russian settlement from Crimean and Nogai raids. Shaped like a reversed letter L, it enclosed roughly 400 by 300 kilometers of black-earth country and included eleven new garrison towns established in 1635-37 and eighteen additional forts built by 1653. It pushed the effective military frontier far south of the earlier Abatis Line.
Russia built successive fortified lines, beginning with the Oka Line fortified along about 250 kilometers of the riverbank by 1533, followed by the Abatis Line roughly 100 kilometers further south, and later the Belgorod Line. A network of ranging patrols supplemented these defenses; by 1623, there were 180 interlocking starozha patrol units, each covering an area with a radius of up to 70 kilometers. Despite these measures, one estimate puts the number of captives taken from Russia between 1600 and 1650 at between 150,000 and 200,000.
The Cossacks were Slavic frontier people organized into two military communities by around 1500: the Zaporozhian Sich on the Dnieper bend and the Don Cossacks on the Don River bend. They raided Crimeans and Nogais, served in Russian and Polish armies, and sometimes rebelled against both powers. The growing number of Don Cossacks on the upper Don also provided Moscow with an additional source of intelligence about steppe movements.
Peter the Great failed to take the Ottoman fortress of Azov in 1695 because he could not control the river and prevent resupply by water. In 1696, he had ships built at Voronezh, sailed them down the Don River, and captured the fort. Russia was forced to return Azov after its defeat in the Russo-Turkish War of 1710-1711, and retook it again in 1736 during the Russo-Turkish War of 1735-1739.
Poland faced structural disadvantages: its core was in the west and it was repeatedly distracted by wars with Sweden; its near-aristocratic republic structure weakened royal authority and made a consistent frontier policy difficult; and its Catholic identity alienated the Orthodox eastern population. Historian Brian L. Davis suggests that Moscow's ultimate advantage was the comparative absence of restraints on its ability to command resources for war, giving its autocracy a more effective military machine.