No, the Russian expansion into Siberia had little resemblance to "the conquering of the Wild West". Maybe in the beginning there were similarities: both "the West" and Siberia consisted of vast lands inhabited by indigenous groups who didn't have an established "state" and thus the lands seemed to be free to conquer. But the difference was that the Russians did not actually need the land, they were more interested in what the land had to offer: the expansion into Siberia mostly followed the fur trade, and of course the need to exert influence further and further. While the expansion into the American West was for the purpose of establishing colonies at first and then a nation: territory was, in fact, the main thing the land had to offer.
Another difference is that the population which carried out the expansion into the American west consisted of free individuals who were able to work for themselves and thus it was easy to appeal to their enthusiasm to improve their lives and shape their own destiny. Meanwhile in Russia a large portion of the population were serfs bound to their landlords. The peasants were emancipated only in 1861, with the serfs working on Imperial lands even having to wait until 1866. Capitalism was just beginning to take shape and for many of these peasants being free was more of a problem rather than a chance to make their own destiny. All those opportunities favored by the American settlers just weren't created yet in Russia.
Russia was not "land of the free" at the time, it was not a nation in the making and it was not a democracy. It was not viewed as a place for fresh opportunities, there were no money or immigrants pouring in from abroad. It was a very different place compared to America.
The expansion into Siberia was hindered in the first place by the Ural mountains and second by the unfriendly locals beyond, in times when Russia was kept rather busy with its European neighbours, too. The first Russian military expedition east is mentioned in the Laurentian Codex, a 14th century document which says that some Guran Rogovich from Novgorod went to the "Ugra lands" in the 11th century. In the following couple of centuries clashes with Mongols and Tatars stopped the Russian drive east, until Ivan the Terrible's successful conquest of Kazan in 1552 proved that there were other viable options beside defence. Some of Russia's neighbours also expressed interest in becoming alleys rather than fighting against it.
Proper expanse was made possible after a route through the Urals was established in 1597. The Cossacks who inhabited the nearby regions and were allied with Russia became the vanguard force of the expansion. The first Russian town in Siberia was Tyumen - annexed by the Cossacks from the Siberian Khanate in 1585. It was an important stop on the trade route between Central Asia and the Volga region.
Sometimes Russia would be asked to protect a local ruler and eventually would gain a bit of land that way. The city of Tomsk, for example, started as a fortress in 1604 in aid of a Tatar noble against Kirghiz bandits. Later on the land was ceded to the Russian Tsar. Accepting Russian rule was often a way to avoid bloodshed between local tribes.
Unlike America, Siberia was not a colony but part of the expanding influence that would turn Russia into a vast Empire. Until the 16th century, Siberia was populated mostly by lots of small indigenous groups who were independent. The Russians made an effort to come to them in peace, integrate them into the administrative domain and make them pay taxes and tributes. Some of them, such as the Chukchi were defiant and fought back vigorously but the battles were not great (a serious defeat at Orlova river cost the Russians only 51 men). With its lack of arable land, Siberia wasn't of particular interest to anybody but it did contain something of value - natural resources and trade routes. It is believed that the original Siberian population counted around 300 000 people spread over a territory of 5 million square miles.
As fur trappers went east, soldiers followed and forts were built to defend the trade. More and more territory was claimed for Russia. Much like in America, Siberian settlements were built with settlers expecting riches coming in. The settlers were hunters, trappers, missionaries and fortune hunters such as craftsmen and merchants; state employers and soldiers but also prisoners and exiles send away for safekeeping, or runaways and outcasts seeking refuge. Siberian furs, ivory from mammoth tusks and goods from China such as silk, porcelain and tea were traded to the aristocracy back home. It was of little use to the common man.
Cooperation of the locals was encouraged by various means: if negotiations didn't work, brutality was applied. Resistance was subdued relatively quickly as trade was rather profitable. For example, obtaining precious furs without damaging them was rather difficult. Locals' skills in the trade were in high demand. The subdued tribes were treated reasonably well if they cooperated, although hostages were often taken in order to assure obedience.
By 1700 most of Siberia was taken in a way that was convenient for both sides - tribute systems were established, Christianity was spreading, trade was booming. By 1709, 230 000 Russians had settled in. The locals were incorporated into the Empire rather than being driven away. Military forts and trading posts would turn into towns and cities, such as Yakutsk or Irkutsk. The furthest regions of Kamchatka were reached in 1697. In the Far East there were some conflicts with China over arable lands but since China was an important market for the Russian furs, a series of treaties were struck to establish the borders between the two.
A big problem for conquering Siberia was the difficulty of establishing transport links. Siberian Route was first major road connecting European Russia to Siberia (and China). It took about a century to complete, starting in 1730. Up to then, transportation was done by river routes, which presented specific challenges such as finding short portages between them. Good roads suitable for wheeled transport remained scarce for quite a long time. The Trans-Siberian Railway, the longest railway line in the world at 5772 miles was constructed between 1891 and 1916. To traverse the same route on horseback would take several months or even a year. America was quite ahead transport-wise, with 9000 miles of railways ready by 1850, with the land grand system giving an additional push, and not to mention the numerous canals and turnpikes which were unheard of in Siberia.
...to be continued