Summer: assuming a normal amount of rain in spring and summer, bring or buy here plenty of mosquito repellant. Those litle buggers can be awful.
Winter: well, we are pretty far north, so bring warm clothes, but you already knew that, right? Mittens/gloves, and a hat.
Camping/Beach/Outdoor Gear: There are a lot of beaches, and a new pool being built right now. There is nice swimming off the pier at the Memorial Union. So bring a suit if you wish.
Winter. There is cross-country skiing in a lot of places here. Skis if you wish.
The weather of Madison fluctuates quite a bit from season to season. The winters are notoriously cold, most winters reaching -20F at least once, while the summers can be hot and incredibly humid. These, of course are the extremes should you be caught in a hot or cold spell. The summers are usually very nice. When visiting in the spring or summer, bring all manners of clothing as the temps can vary significantly from day to day or even from hour to hour. The spring will sometimes see heavy afternoon thunderstorms, especially on very humid days.
Miscellaneous: Don't forget to bring your spare liver as people like to party pretty hard here.
Clothing/Shoes/Weather Gear: If you're unfortunate enough to travel to Madison in the dead of winter during January till March then make sure you bring lots of heavy winter clothing, including gloves, hats/ear muffs, and boots. Early Spring and Late Fall present opportunities for period of rain showers, raincoats and umbrellas are a must if you want to keep yourself dry. During early summer, there are opportunities for thunderous thunderstorms during the afternoons so make sure you have an umbrella ready. The rest of the summer, the weather can be extra hot and humid, so light cotton clothing, shorts, and comfy footwear are recommended.