During what time period did Russia wrestle control of lands that belonged to the Khanates and other tribes that occupied its borders? What were the battles and wars like? Is that time period looked at by modern Russians in a similar romantic way that America views the Indian Wars?
Also, are there any books to recommend on the subject?
There are a lot of parallels with America's Westward Expansion!
If you're interested in Russia's Eastward Expansion, the book Glorious Misadventures is all about it. Long story short, the first phase of the expansion was driven in large part by Cassocks, who with and without permission from the crown went to war with the Sibirs and other tribes, and pushed Russia's boundary east one major river system at a time. You can probably draw some parallels between them and the Mormons pushing west in America, or with the US Army and the forts they constructed to control the territory, as the Cassocks did. Just like how a lot of America's westward expansion was driven by immigrants from other countries, a lot of the Cassocks came from Poland looking for land of their own.
The boundary lands, like the American Wild West had low control by the central government. So much so the provincial governor was recalled by Peter the Great and executed for coining his own money (with his own face on it) and otherwise turning it into his own little kingdom.
Instead of a gold rush, Siberia had a "fur rush" with the courts of Europe spending ludicrous amounts of money for Russian furs. So many animals were trapped that they progressively went extinct in a line moving eastward, as the Russian adventurers moved east. Around 1810, just one fur company was responsible for 10% of Russia's entire GDP.
Rezanov, the son in law of the owner of this company, successfully lobbied the tsar to conduct a trade mission to Japan and explore the Pacific Northwest along the way. He wanted to push into California to continue the rather rapacious and exploitative fur trade on the Pacific side of things, since the Chinese market for furs was exploding - and he wanted to open Japan for trade as well.
His mission was a mixed success (hence the title of the book), with him utterly failing to open Japan to trade - he was quite possibly the worst possible ambassador - but managing to make contact with the Spanish in San Francisco, and exploring Oregon at the exact same time Lewis and Clark were there!
He got the Spanish governor to ignore, somewhat, the prohibition on dealing with the Russians, and plotted to conquer the whole place. He seduced the governor's daughter (which Russia's number one rock opera for 30 years, Juno and Avos, is about) and agreed to travel back to Russia then Italy to ask the Pope to get a dispensation to marry her.
He died travelling back through Siberia, and the Russian dream of a Russian America died with him. Alaska was eventually sold to America a couple decades later, and the Russian forts in America (like Fort Ross) are now just historical curiosities of a possibility that never materialized. The governor's beautiful daughter didn't hear about his death for years, and chose to never marry when she did find out. She lived well into the Gold Rush era as a nun in Monterey.
Due to the extreme lucrativeness of the fur trade, a lot of world powers had their eyes on the Pacific Northwest during this time period, but the political and military maneuverings are sadly almost entirely forgotten.
Your question is a little problematic in that it is rather broad. The Russian imperial expansion into Central Asia and the Caucasus lasted for centuries (roughly 1500-1800s). Imperial policies would often vary quite widely over time. For example, Catherine the Great often treated the elite of conquered lands as servitors to the Russian Empire. She allowed a degree of local rule in exchange for loyalty. This was in stark contrast to the Central Asian campaigns and settlements of the mid-19th century in which tried a half-crazy attempt to bring in Russian colonization of Central Asia and an integration with the Russian metropole. In short, the Russian Empire's expansion was not a homogenous process.
Some good books on the subject:
Hosking, Geoffrey A. Russia: People and Empire, 1552-1917. Cambridge, Mass: Harvard University Press, 1997
Sunderland, Willard. Taming the Wild Field: Colonization and Empire on the Russian Steppe. Ithaca: Cornell University Press, 2004.
Slezkine, Yuri. Arctic Mirrors: Russia and the Small Peoples of the North. Ithaca: Cornell University Press, 1994.
Russia's Steppe Frontier by Khodarkovsky is a truly excellent book on the subject, and I would not hesitate to say it is the absolute best. I can't do its nuance justice with a summary here, but I asked Dr. Khodarkovsky the same question a few years ago, and he said that it was similar, but not the same.
More importantly than the superficial reasons, to him at least, was that the steppe provided a sort of uniting impetus. Outlaws and dissidents often went cossack to escape settled society, and raids not only made the border dependent on the central authority of Muscovy and later Moscow but distinguished civilized people, Russians, from uncivilized people. The nomads either were vassal to the Russian state or they were outlaws.
On the other hand, America was very much interested in civilizing the Indian, seeing it as their duty. Russia was not making peace treaties or trading beads with the steppe. It was also, in Siberian cases particularly, not interested in settling this land, with a strong distinction to American manifest destiny. His comparison was much more to European colonization - it had a certain disinterest in the natives, with the exception of utilizing them to harvest resources.
The Russian conflicts with the Ilkhanate and the Crimean Khanate are very, very different than their conflicts with the steppe. Those were both settled societies and polities, while the steppe are not.