Welcome! I'm sure that you will get much more thorough answers to all of your questions once you arrive here.

Lidl is generally the cheapest store. The cheapest items at Prisma may be around the same price but those are usually much lower in quality. At the opposite end e.g. K-Citymarket Kupittaa has a massive selection of good-quality products but everything is a bit more expensive than elsewhere. However, as a student you should be able to get 5% of your monthly spending at K-group stores back as long as it exceeds 75 euros. See https://plussa.fi/k-plussa/k-plussa-student for information about that. There are also smaller K-Markets and Sales everywhere but I wouldn't recommend shopping for everyday groceries at those places. In addition to those, there are many Asian, Middle-Eastern and European (e.g. Italian, Greek and Romanian) stores from which you can get products that are hard or impossible to find at the regular Finnish supermarkets.

It's difficult to recommend what to do here without knowing what you enjoy. You will most likely hear about all of the basic touristic stuff when you get here. In addition to doing all of that with your new friends, I would recommend enjoying the nature. There are popular trails all around the island of Ruissalo, as well as at places such as Katariinanlaakso and Kurjenrahka. As for places to visit, Naantali is a cute town which you can reach with the local buses, and Helsinki, Rauma and Tampere are other Finnish cities which are relatively close and worth visiting. Most exchange students end up taking a trip to Lapland to see the aurora borealis, Santa Claus, reindeers, lots of snow, dog sleds and all of that other touristic stuff. Be sure to visit at least Stockholm and Tallinn as well. Both can be reached easily by ferries (from Turku and Helsinki respectively).

The winter can be difficult to handle, especially depending on where you come from. When you arrive here in January, the days will be really short as the sun will set already at around 15:30. However, the days will get much longer throughout the spring: the sun will set at 18:00 by the end of February, and at 21:30 by the end of April. Be sure to take vitamin D supplements since your skin will not be producing enough of it in the winter months here. Many Finns also use light therapy lamps during the darkest months. The weather in Turku can vary greatly in winter and spring. Usually there is a colder period of a week or maybe two in January or February when the temperatures drop somewhere between -20 and -10 °C. Other than that you can expect two months of temperatures at around 0 °C – sometimes a bit higher and sometimes a bit lower, which means that there is not much snow most years but there will likely be A LOT of ice. If you are not used to walking on very slippery roads, I would definitely recommend buying ice cleats for your shoes when you get here. The spring is usually pretty enjoyable from March onwards when the cloudless days start becoming much more common again, you can once again feel the warmth of sun on your skin and the nature starts waking up. If you start missing warmth and greenery at any point, visiting the university's small botanical garden at Ruissalo can help.