First of all, the majority of the Finnish army was demobilised at the end of the Continuation War, as per the treaty with the Soviets, so it was a minority of the Finnish army that fought against the Germans.

The Finns informed the Germans by the 2nd of September 1944 that they had to leave Finland by the 15th since they had signed a separate cease fire. The Germans evacuated all their assets in souther Finland in Operation Birke, but could not evacuate the 20. Gebirgsarmee in northern Finland in time - on the 21st of September the last German convoy left Finland, and the German troops left had no choice but to retreat to northern Norway.

Initially, both sides had a secret agreement where the Germans would inform the Finns of their planned retreat of that day, do only superficial damage to the infrastructure, leave everything else in order and then retreat, while the Finns would advance over the same terrain just after the Germans had left. Both sides also cooperated in evacuating the population of Finnish Lapland except the people in and around Tornio - in all some 168 000 people were evacuated southwards or to Sweden before the outbreak of serious hostilities.

However, the Soviets caught wind of this 'gentlemens' agreement' and initially started pressuring the Finns to advance further, and then offered Red Army 'help' to push the Germans out of Finland (ie Red Army troops permanently stationed in Finland). In order to avoid this, the Finns started a real advance and landed in the German rear at Tornio on the 30th of September 1944.

Before this, the Germans tried to sieze the island of Hogland in the Bay of Finland on the 15th of September but failed, losing more than 1 300 men in the process.

On the 28th of September, the secret agreement had broken down between the 20. Gebirgsarmee and the Finnish forces pursuing it, causing fighting to break out between advancing Finnish forces and German rear guards.

As a retribution for the Finnish hostility, the Germans started torching all buildings they left and tore up all infrastructure completely. The town of Rovaniemi was almost completely destroyed by the Germans.

The Finns attempted several flanking attacks through the forests, but it seems like there was a general lack of enthusiasm over fighting the Germans, which combined with war-weariness, the low number of Finnish troops (the Germans had about 210 000 men, the Finns about 75 000) and on several occasions left behind stores of alcohol (which the Finns stopped to consume) prevented these actions from surrounding any larger amounts of German troops.

So, to answer your question - from the start the Finns did not want to fight the Germans and made a secret agreement with the Germans to advance after the Germans had left a piece of land. After Soviet pressure, the Finns made real attacks on the German forces. The Germans then torched large parts of Finnish Lapland, which soured the Finns towards the Germans. Yet there were still a general lack of enthusiasm for fighting the Germans both from the soldiers and the officers. All knew that the Germans were heading for Norway and would leave in a few months anyway - the Finns had been at war since 1941 and many were war-weary and wary of risking anything for something that could be achieved without combat, but had to attack due to Soviet pressure.