The main involvement the Nazis had with Finland during the Winter War was blocking military aid intended for Finland (e.g., from Italy). The Molotov-Ribbentrop Pact (signed a week before Germany invaded Poland) prohibited Germany and the Soviet Union from aiding an enemy of the other, so German aid to Finland was prohibited. Allowing military aid to pass through German to Finland could quite reasonably and readily be interpreted as aiding Finland, so the Nazis blocked such aid. The secret part of the Molotov-Ribbentrop Pact also defined German and Soviet spheres of influence; Finland was within the Soviet sphere.

After the Winter War, Germany and Finland resumed relationships - which had been traditionally friendly - and both countries were involved in planning for war on the Soviet Union: Operation Barbarossa for Germany, and the beginning of the Continuation War for Finland. Despite this planning, the Finnish government claimed to not have been involved, and presented itself as a "co-belligerent" with Germany, and not an ally. During the Continuation War, German troops were fighting in Finland, both as part of the (unsuccessful) German attack from Norway to take Murmansk, and as support for Finland (tactical air support, an infantry division, and an assault gun brigade). Germany also provided military supplies, some of which were critical to the Finnish effort, such as large numbers of anti-tank rockets (Panzerfaust and Panzerschreck) just in time to help Finland stop the Soviet advance at Tali-Ihantala in July 1944.

With a peace agreement between Finland and the Soviet Union ending the Continuation War (with the ceasefire on 4th September), Finland went to war with Germany, in order to expel German troops from northern Finland, as required by the peace agreement.

See https://www.reddit.com/r/AskHistorians/comments/c6xyyd/why_did_the_finns_join_the_axis_powers_in_wwii/ for more details on Finnish-German relations (and references).