Do I have to wear a swimsuit?
Well...there is always a chance of falling in. After all, SUP is a water sport. I like to wear a swim suit under an outfit that I would typically wear for a short run or workout at the gym. If you don't wear a swim suit, it might be a good idea to pack an extra outfit...just incase you fall off of the board. I would recommend against wet suits once the water is warm enough to swim in because the wet suits get really hot when you are on top of the paddle board. Early spring or fall would be better times for using wet suits.
Some of the morning fitness classes on the paddle boards have been cool. If it is cool out, it is a good idea to layer (with a light sweatshirt or long sleeve shirt). Wear something that you don't mind getting wet or tying around your waist. On cool mornings, some participants like to wear their lifejacket to stay a little warmer (and heck, it isn't a bad idea to have that PFD close by anyway).
Bottom line: what you wear out paddling is up to you. Do what you can to get out and enjoy Minnesota's beautiful summer weather on the lake, and have some fun!