Trying to stay as neutral as possible. Obviously with Soviet Union being so close and all, they had a big say in Finlands doings, and even though Finland was (if I recall correctly) offered help via the Marshall Plan, Finlands leaders rejected it because of fear of Soviet Union.

Almost throughout the whole Cold War Finland had good relations to both the Soviet Union and to the West. A big booster to that was Urho Kekkonen, who managed to get the Soviet leadership like and trust Finland.

Soviet Union did trust Finland so much that they even gave us back "Porkkala" which was leased to SU during 1950's, even though the area was leased for 50 years after WWII.

There were some minor crises between Finland and Soviet Union during the 1950's and '60's, but none of those ever escalated much. One of the crises was related to the Berlin crisis of the 1960's, which demanded Finland to basically be in a military alliance with the SU. At the same time the note was given to Finland, Kekkonen was having a holiday in the USA, and Kekkonen didn't even stop his vacation early to deal with the crisis. The crisis led into the breaking up of the Finnish Parliament and for new elections to be placed in a couple of months.

When the cold war ended though, Finland immediately started rushing into the west, and was one of the first countries to join the European Union in 1995.

tl;dr: Finland was trying to stay as neutral as possible, while pleasing Soviet Union at the same time.

Sources: http://www.otava.fi/oppimateriaalit/lukio/forum/ Forum IV, which is a finnish history textbook. Also my history teacher

The cornerstone of Soviet-Finnish relations was the Agreement of Friendship, Cooperation, and Mutual Assistance signed with the USSR in April 1948. The Treaty guaranteed that Finland would aid the Soviets against "Germany or its allies" and fostered a series of networks and political connections between the Soviets and the Finns. The Finnish political leadership, exemplified by Juho Paasikivi Urho Kekkonen, rationalized that the West could not help the Finns against the Red Army and their best option would be to adapt a strategy of accommodation. This Realpolitik was born out of their experience in the Second World War; Finland had twice taken on the Soviets (the Winter War and WWII) and bore the brunt of the fighting. This leadership gambled that if the Finns gave the Soviets concessions on security issues, they would have more latitude in domestic and international relations. Max Jakobson, the Finnish ambassador to the UN summed up the postwar Finnish perspective thusly:

the Finns have usually had no more than marginal control over the external circumstances of their country; their freedom to choose between two evils. But ultimately it has been their own decisions, not decisions imposed by others, that have determined their fate.

The Finnish political leadership managed to employ the Treaty as a sort of political jiujitsu in their dealings with the Soviets. The fact that Finland had a treaty with the USSR meant that Finnish leaders could engage in "strategic foot-dragging" when the Soviets wanted to alter or change Finnish behavior. Kekkonen proved pretty adept at deflecting Soviet concerns during his long tenure as President. He was able to capitalize upon his personal connection with Khrushchev and other Soviet officials, including the KGB. For example, when the Soviets were concerned that Finland's joining the Nordic Council was a sign that Finland was seeking closer political ties with the West. Kekonnen negotiated a quid pro quo with the KGB and then Khrushchev that would allow Finnish membership in exchange for Finnish diplomatic recognition of East Germany. He was thus able to present Finland's presence in the Nordic Council as congruent with the larger geopolitical strategy of the USSR and fully in compliance with the Treaty. Later in the 1960s, he would advocate that Scandinavia should become a nuclear-free zone. While this was something of a dead-letter among NATO, it made a neutral Finland appealing to the Soviets because it offered the promise of detaching Scandinavia from NATO.

One of the unintended consequences of the Treaty was that it created a degree of consensus and stability within Finnish politics. Because the Soviets saw Paasikivi and Kekkonen as political allies, they ordered the Finnish Communist Party to cooperate with the government. When the Finnish government discovered to their dismay that Finland was less than the pliable tool they pretended to be, the Finnish political establishment had deprived the local Communist Party of much of the economic and social issues which they could agitate against. Moreover, the Treaty called for the elimination of anti-communist groups and classroom materials within Finland, further depriving the Communist Party of an issue. By keeping to letter of the Treaty, the Finnish political establishment enjoyed a degree of legitimacy within the Finnish left. This paid dividends during 1968, when the Soviet invasion of Czechoslovakia prompted a split within the Finnish Communist Party with the majority adapting a nonaligned, almost Eurocommunist, line.

Although Finlandization was a pejorative aspersion in the West during the Cold War, it's not really deserved as the Finnish political leadership enjoyed a tremendous dexterity during this period. Finnish leaders were quite realistic about the strength of the Soviets and their willingness to squash independence in the name of Soviet security. By using patently pro-Soviet Treaty to preserve Finnish independence is a remarkable feat of postwar diplomacy.

Sources

Jakobson, Max. Finnish Neutrality; A Study of Finnish Foreign Policy Since the Second World War. New York: Praeger, 1969.

Jussila, Osmo, Seppo Hentilä, and Jukka Nevakivi. From Grand Duchy to a Modern State: a Political History of Finland since 1809. London: Hurst & Company, 1999.

Luostarinen, Heikki. "Finnish Russophobia: The story of an enemy image." Journal of Peace Research 26, no. 2 (1989): 123-137.

Rentola, Kimmo. "From half-adversary to half-ally: Finland in Soviet policy, 1953-58." Cold War History 1, no. 1 (2000): 75-102.

It should be noted that Finland joined the Nordic Council 1955, allying themselves with the west in at least one official way.