I think that there was a strong strand of distrust in the government at the start of WWI. The Russian Revolution of 1905, the failures of Russia in the Russo- Japanese War, Bloody Sunday, and the perceived distrust of the October Manifesto's goals all had contributed in a very predominant loss of faith in the Czar. Unwilling or unable to fully grasp how the political dimensions had changed, the Czar believed that waging war would be a strong way to mobilize individuals in the cause of patriotism and support of the nation. Russian casualties in the war began to mount with objectives being unclear and faith in the ruling order invisible. Such a lack of faith was seen in troops that refused to carry out orders being so poorly equipped and convinced of the war as being an error. This contributed to the overthrow of the Czar, reflecting the Russian dissatisfaction at the start of the war manifesting itself into outward rebellion.
The truth is that how the Russian people felt about anything was completely irrelevant, the Tsar was an absolute autocrat. Although the Duma was more or less a parliament of sorts, they had no actual power over any decision the ruler made. Many Russians supported the war as an act of solidarity with fellow Slavs in the Balkans, but these would have been military officers and politicians. The people of Russia as a whole had little or no idea of what was going on in the world outside of their locality, the great mass of Russians being uneducated people living in rural areas, for all practical purposes cut off from the rest of the world. Those in the cities were more informed, but largely opposed to the war. People were often supportive of war against Austria-Hungary, but they feared war with Germany (with good reason).
The military command structure supported the war, for personal and professional reasons, although Grand Duke Nicholas (the tsar's brother, who was made commander-in-chief of the army) thought it would be a disaster for the country and likely lead to revolution. It was said that he cried upon being informed of his appointment.
The Russian war plan was foolish in the extreme. To relieve pressure on the French the Russians intended to invade East Prussia before their own mobilization was complete, thereby hoping to throw the German war machine out of balance. Operations against Austria went well, since the Russians had (through their effective use of spies) the complete Austrian war plans years in advance. Against Germany, the Russians foundered. They sent two Armies into Prussia, one south and one north of the Masurian Lakes. These two (the First and Second Armies) were to coordinate somehow, with no possible means of communication, and with both commanders hating one another. Under a retired Prussian general recalled to duty (Paul von Benekendorff und Hindeburg) and a logistical wizard named Hoffman the Germans moved their forces by lateral railways and crushed one Russian army at Gumbinnen and the other at Tannenberg in a matter of days. From this point on the Russians were fighting a losing war against Germany, which eventually cost the Tsar his throne in spring, 1917, and then the Kerensky government to fall to the Bolsheviks later in the year.
By the end of WWI (actually before the end of the war), Russia would pull out of the war and the government would fall, first to Kerensky's people and then to the communists. But this is not something that you would have predicted at the beginning of the war.
Instead, there was a great deal of support for the war among the Russian people at first. This was largely for ethnic reasons. The Russians, who were Slavs, felt kinship with the Slavs of the Balkans. These Slavs were ruled by the ethnically German Austria-Hungarian Empire.
So, out of ethnic solidarity, the Russian people were fairly eager to go to war with Austria-Hungary and even with Germany.