Try to not do your apartment searching in Winona during the winter. There just aren't as many people moving out of their homes at this season in Minnesota. As a result, if you try to shop at this time of year you will find that you have very limited options for new apartments.
Landlords take meeting you in person pretty seriously in Winona. Most people will be able to afford the low rents here, so the decision to give someone an apartment becomes about trusting that the tenant will be responsible and look after the unit. It can be difficult to convey that to a landlord if you only meet over the telephone or Internet.
Winona does operate a small bus service. Rides only cost a dollar, but the lines shut down after 6:00 PM, and there is no service on Sundays. If you work nights or weekends, you will probably want to have a car. There is a lot of great countryside to explore outside Winona, and a car also gives you the freedom to do so much more readily.
Getting out of the city to explore the lakes is essential. Boating, hiking, biking and cross-country skiing in the winter are all popular activities. There are also several beaches you can visit in the summer. Downtown is a very scenic historic district where you can shop and grab a bite of lunch. Winona is also a very bar-friendly city, with loads of options. Ed's No Name Bar is a good starting point, with a good selection and a summer beer garden.