Phew, this is a fairly big question...

I have to be brief, but here goes: The key problem for the government was that the public was adamantly against the Soviet territorial demands. No cabinet member wished to be seen as "Russkie" appeaser. Of those in government, only Social Democrat finance minister Väinö Tanner and minister plenipotentiary J. K. Paasikivi - another veteran politician and the leader of delegation negotiating in Moscow - were in favour of giving in to Soviet demands, at least to some extent. These two were supported by field marshal Mannerheim and some other high-ranking officers, who had a dim view of Finnish capability to resist Soviet aggression.

The rest seemed to believe the Soviets were only bluffing, or at least there would be another round of negotiations. In late November there were already calls to demobilise the army, and despite the Mainila shelling incident on 26th November and the following Soviet repudiation of the non-aggression pact, the attack on the early morning of 30th came as rude surprise to many. The reaction of prime minister Cajander is legendary: several memoirs report that after receiving news of the attack, he first asked the minister of the interior Urho Kekkonen whether it was really necessary for the cabinet to assemble before office hours.

Prior to war, the Finnish foreign policy had been basically based on proclamations of neutrality and the expectations that in the event of war, the League of Nations and fellow Nordic countries would somehow resolve to come to its aid. The Soviet Union was viewed with suspicion at best, downright hostility at worst, and it really seems that many did not actually understand that the Soviet Union might really be concerned about the possibility of Finland's territory being used as staging area for attacks against the Soviet Union. The ones who understood were military men, Mannerheim in particular (who had been a general in the Imperial Russian army and understood perfectly what defending St. Petersburg/Leningrad required) and some more astute politicians. The rest of the politicians were, I believe it is fair to say, quite provincial characters who frankly lacked imagination and understanding of foreign affairs. Furthermore, there were only hopes that foreign powers would come to Finland's aid somehow; very little concrete planning had been done, except with Estonia for blockading the Gulf of Finland.

The hopes that the Nordic countries at least would come to Finland's aid might have held some water had the war been an isolated attack by Soviet Union against Finland; but in the geopolitical situation of late 1939, everything had changed. There were of course hopes of Western aid, and these hopes were raised whenever foreign statements of support were proclaimed. But these hopes were for all intents and purposes dashed when the Swedes denied the Western allies access through its territory. This was probably an adroit move, since it seems very likely most of the Western "aid" would have remained in Northern Sweden to secure the Kiruna iron ore fields in order to prevent shipments to Germany.

Nevertheless, up until the end there were even cabinet ministers who refused to give in to Soviet demands. Defence minister Juho Niukkanen from the Agrarian Party was from eastern Viipuri region, an area that was included in the territorial demands, and resisted peace to the last. Similarly, Agrarian Party minister of education Uuno Hannula resisted even negotiations. Both resigned soon after peace of Moscow was signed.

As for the fears, hopes and expectations of the general public, these were more diverse. It might be a good summary to say that most feared a war would break out but somehow expected it wouldn't happen; but if it happened, they would fight. There seems to have been certain fatalistic air among the populace, exemplified by Risto Ryti, then the governor of Bank of Finland: "We shall brace ourselves for most honourable annihilation."

After all, it was fairly well known what had happened to small minorities (including related Karelians) in the Soviet Union, and many expected that should the country be occupied, they would be tortured, executed or packed to cattle wagons headed towards Siberia. There is one story, possibly apocryphal, of a farmer in Eastern Finland who during the early days of the war heard from the radio that the Soviets had crossed the Terijoki river. The announcer meant a river in the Karelian Isthmus, close to Leningrad, but a river by the same name ran only 20 kilometers from the farmer's home. Exclaiming in horror that the "russkies" are soon here, he immediately shot himself.

It is well documented that the terms of the armistice and Peace of Moscow were extremely hard for many to accept: even in the front lines, there were many who said they'd have continued to fight had they known the terms. These are strong words from soldiers who had fought without pause for weeks, in companies reduced to less than 20 men in places.

That said, it is of course clear that there were some who even welcomed the Soviet attack, or couldn't care less about what happened. The Communist party had been outlawed in the 1930s, but there were still underground activists, and in poor regions in Eastern Finland in particular, there were even instances of people hailing the Red Army as liberators. However, these were always in the minority, and even a significant number of communists actually fought against the Red Army. There was even a fairly large group of "Red" veterans of the Civil War who asked whether they could fight somewhere, despite having no military training besides their "practical experience" 21 years earlier.

Sources:

Key source is an excellent book discussing these questions and more,

Tepora, T. (2015). Sodan henki: Kaunis ja ruma talvisota. Helsinki: WSOY.