The Continuation War was very much started by Finland, as far as it can be considered separate from the German invasion of the Soviet Union. If the Continuation War is considered as part of the broader war on the Eastern Front, then the answer would be that Germany started it (without the German invasion, Finland would not have gone to war against the Soviet Union).

How did it begin? After the Winter War, and the invasion of Norway by Germany, and the collapse and surrender of France, Finland was isolated from Britain. Britain and France had promised aid, including troops, during the Winter War, and these offers appear to have made the Soviets more willing to agree to an acceptable peace (since they really didn't want war with Britain and France). France was out of the picture, and with Norway in German hands, there was no feasible way for Britain to assist Finland, either against Germany or the Soviet Union. Finland was caught almost-between two powerful and dangerous giants: Germany and the Soviet Union. The Soviet devouring and digestion of the Baltic states (Estonia, Latvia, and Lithuania) showed Finland its potential fate if the Soviet Union made another attempt to conquer them.

Relations with Germany had cooled greatly due to the Winter War - Germany honoured the non-aggression pact with the Soviet Union, including the secret protocols placing Finland in the Soviet sphere of influence, by refusing to provide military aid and also by blocking military aid from passing through Germany (thus, aid sent by anti-communist Italy didn't reach Finland due to Germany). Germany had been a friend in the past, having sent troops to help the Whites during the Finnish Civil War (since the Whites won, this was a "friendly" act). Good-will between the Finnish government (if not the defeated Finnish Reds) and Germany was such that a German prince was chosen to become the King of Finland (with the end of WWI, the king-to-be abdicated before his coronation, and Finland became a republic instead).

The Winter War was followed by normalisation of relations between Germany and Finland. As they grew closer together, Finland was interested in the possibility of German assistance in case of future Soviet attacks. Finland was also interested in avoiding being an enemy of Germany, with the accompanying prospect of German invasion. Both sides benefited from closer relations. Finland gained short-term safety, and Germany gained transit rights to support their armed forces in northern Norway. With military cooperation in place (the transit of Finland by German troops), Germany began to explore the possibility of Finnish assistance for an attack on the Soviet Union.

The Winter War had shown that Finland's army was unable to cover the long frontier - the northern end of the border had been only minimally defended during the Winter War, with the weather doing far more to stop the Soviet advance in the north than the few Finnish troops in the area. The Finnish army also had no realistic chance of taking Murmansk. Northern Finland would be a German zone of operations, with German troops from northern Norway attacking towards Murmansk. This needed Finnish permission. More generally, Finnish support for Barbarossa was desired - the more the Finns did on their front, the less Germany would need to do. Thus, Finland was deeply involved in the planning for Barbarossa.

Operation Barbarossa began early in the morning on 22nd June 1941. Finland's initial moves were restrained: landing troops in the demilitarised-by-treaty Åland Islands, and arresting Soviet consular staff there. Finland also began laying mines in the Baltic. The immediate Soviet response included air raids on Finland, using a small handful of aircraft. Stalin ordered Soviet forces to counterattack Axis forces that had crossed the border, and to bomb German forces on the German side of the border; these air raids appear to have been part of that response. A few hours later, at about 9am, Stalin ordered a general counter-offensive along the entire front (not a wise, realistic or implementable order). The Finnish front remained quiet until 25th June, when major air attacks were launched by the Soviets, using about 460 aircraft. The main targets were airfields being used by Germany, but civilian targets were hit as well, with Helsinki and 5 other Finnish towns being bombed. Finland's response was to declare war on the Soviet Union, and Finnish troops moved to attack a few days later.

Noting that Finland's remilitarisation of the Åland Islands broke the peace treaty with the Soviet Union, and allowing Germany to use airfields in Finland for attacks on the Soviet Union was an overt act of war, the Soviet Union would have been justified in declaring war on Finland. This would have allowed Finland to paint them as the aggressor, forcing them into war, and perhaps the air raids on the airfields took place without a declaration of war in the hope that Finland would stay out of the war. However, Finland's response was to declare war on the Soviet Union, using the Soviet air raids as justification and presenting themselves as peace-loving victims of Soviet aggression in both domestic and international propaganda. Finland was responsive to international opinion, and while not willing to stop at their pre-Winter War borders, they did halt their advance short of cutting the Murmansk railway, and stop far enough from Leningrad to claim that they took no part in the siege (Finnish troops were part of the force surrounding Leningrad, but despite German requests, they refused to actually attack Leningrad). The Finnish halt resulted in a relatively friendly declaration of war by Britain (along with what was basically an apology for the declaration of war, and an explanation that it was to make the Soviets happy) followed by no offensive British action, and no declaration of war by the USA. Finland presented itself to the world as a "co-belligerent" of Germany, and not an ally. While Finland had been involved in the planning for Barbarossa and allowed Germany to attack the Soviet Union from Finnish territory (which would to most people qualify them as an ally), they were willing to put their own political goals ahead of German military goals (so they weren't the best or most cooperative ally).

Finland could have avoided taking part in Germany's war against the Soviet Union. First, they could have refused early in the planning stage. Second, even if they were involved in planning, they could have refused Germany the use of Finnish territory, simply agreeing to join in if and only if the Soviet Union attacked Finland. Of course, lack of cooperation with Germany might have led to war with Germany.