I'm a Finn, and in Finland we are of course taught about the history of Sweden and Finland, but it's very much from the Finnish perspective, focusing what happened in Finland, how Sweden was from time to time in war with Russia in modern Finnish territories and so on until losing Finland to Russia in 1809.
Less so did I learn on what was the standing of Finland and Finns within the Swedish realm compared to other regions of Sweden and other subjects of the crown. Turku/Åbo for example is the oldest city in Finland and quite close to Stockholm, and for most of Finnish history Turku was the largest city in Finland. What role did it have in the Swedish realm?
Extreme right-wingers and nationalist here in Finland love to try to portray that Finland and Finns were oppressed and Finland was exploited for the good of Swedes. AFAIK that's blatantly wrong, and rather Finland and Finns were in a rather similar position as any other part of Sweden at that time. But I have no detailed knowledge of that time.
Before 1809, Finland was a full part of the Swedish crown. This meant that the Finns, wether Swedish-speaking or Finnish-speaking were full citizens with the same rights and duties as Swedes in Sweden had towards the Crown. Finns were sent to the Riksdag, the estates parliament, Finns served to a large extent in the indelta (alltoment system) army.
It is true as /u/AshkenazeeYankee says, that the overwhelming majority of the nobility in Finland spoke Swedish (or French, as was common among the nobility in the 18th century) as their first language and that they had Swedish roots. However, the Finnish self-owning farmers owned about 90% of the arable land in Finland from the 1300s to 1809 (the same number in Sweden was 33-55% during that era). The Swedish-speaking nobility in Finland did not own a lot of land nor did they have large estates - they came to Finland to serve as officers, officials and other state positions.
The Swedish Crown was generally tolerant rgarding languages during the 17th and 18th centuries. The Riksdag established a permanent translation office in 1735, giving the Finnish speakers the right to adress the Riksdag in Finnish should they so desire. During the 17th and 18th centuries, it was common that the nobility spoke French (and usually their local language), the priests latin (and the language of their parish), the burghers Dutch or Low German (and sometimes a local language) and the peasants spoke the local language (Swedish, Finnish, German, Russian, Estonian, Latvian, Danish, Norwegian). As long as you could communicate with your army officer, police chief and tax collector, the Crown cared little what other language you spoke. Those officials would often be bi- or even trilingual in order to ease communication.
As /u/AshkenazeeYankee says, the push to convert the Finns that believed in old animist faiths or orthodox christianity and increase the knowledge of lutheranism had priests in Finland translate the bible, Luther's catachesis and teach people to read (but not write, it was considered two separate things back then) Finnish. They did the same in Estonia and Latvia when those regions were part of Sweden.
Were the Finns opressed? I'd say no. They had representation on equal term as the Swedes and the same rights and duties.
Was the Finnish language supressed? Not actively. Sure, you had much better career prospects if you spoke Swedish - there were plenty of career oppurtunities for bilingual people. The Swedish crown did not force Finns to speak Swedish.
Åbo was one of the most powerful fortresses in Medieval and Early Modern Sweden (with Stockholm, Älvsborg, Kalmar and Viborg) and was certainly a very important part of the realm. If you read the intrigues of the Wasa sons over the Swedsih throne, you will see how Åbo figures several times as a very important place to hold. It was the capital of the eastern part of the realm and much of the tax collection from the Finnish peasants came through here.
I'll close with a joke regarding Finnish and Swedish relation.
"Sweden always fought bravely to the last Finn." (referencing that the Finns proportionally made up a larger part of the army and the lack of participation in war on Sweden's part after 1809).
"Yes, and Finland always fought bravely to the last Swede." (referencing Marshal Mannerheim, who had Swedish as his primary language and spoke Finnish badly, but still is the national hero of Finland).
The best book I've read that talks about this at all The Northern Wars. War, State and Society in Northeastern Europe 1558-1721*, by Robert I. Frost.
From my admittedly limited understanding, the Swedish Empire of the 17th century seems to have largely evaluated ethnic Finns though the lens of Lutheranism. The ethnic and linguistic characteristics of Finns were considered salient only as they related to the power of the Lutheran Church. Thus, the Finnish language received some limited state support, but only because it made it easier to get Finns to learn the Lutheran Catechism. There seems to have been a deliberate effort by the Swedish state in 17th century to eliminate both pagan and Orthodox Christian beliefs from among the Finnish peasantry. This was not entirely successful, but as you observe, for a long time the Bishop of Turku was the most powerful and influential Lutheran position in Finland. The status of Finnish can be see in the history of the Royal Academy of Abo: In 1642, clergy associated with the University helped published the first full-bible translation in Finnish. At the same time, classes at the University and it's associated Latin school were largely entirely in Swedish, and Finnish continued to be seen as a "peasant's tongue". The languages of culture and church were Latin, Swedish, and German, in more or less that order.
The Swedish crown tended to use Finland as a area to reward large but agriculturally unproductive tracts of land to Swedish or German nobles in reward for their service to Sweden in it's wars. Thus the Finnish nobility (especially the upper nobility, Baron, etc) tended to culturally see themselves as Swedes in one form or another.