What is the day to day language in a place like Turku/Åbo where the official language is Swedish? When I visit there as a tourist (so in shops restaurants etc.) should I expect almost everybody to be speaking Swedish, like Québécois in Quebec?
Swedish is an official language along with Finnish at the national level, and in most coastal areas on the municipal level.
Turku is an officially bilingual municipality, with a majority of native Finnish speakers. Although everyone in Finland (even in officially monolingual areas) learns the other language in school, proficiency in Swedish among native Finnish speakers is usually poor due to bad teaching and lack of exposure and practice (particularly outside of officially bilingual areas), so English is a better bet anywhere except where native Swedish-speakers make up a majority (which does not include any of the major cities).
Speaking Swedish in most of Finland thus only makes any sense if it's your native language (I'm Swedish, and in Helsinki and Turku it really is hit-or-miss as to who will help me out in Swedish and who will reply in English); on the other hand, knowing it will help you read official signs (public transport etc.) in officially bilingual areas.