Wear something that you like and that’s comfortable to move in. Many dancers are into vintage fashion and enjoy dressing up for dances, but lots of people dress casually. In general, dance with a live band are dressier, but many people wear jeans & t shirts too.
Swing dancing is energetic, and chances are, you’re going to get sweaty. Bring a few extra shirts and change as soon as your shirt becomes noticeably damp. This isn't just for men - women often need to bring a change of clothes too. Consider wearing an undershirt to put an extra buffer between your sweat and your partner. Also: wear deodorant, bring mints or gum, and avoid strong perfume.
You'll most likely want to avoid backless dresses, strapless dresses, extra high heels/stilettos (or heels at all if you’re new to dancing – try a low-tread sneaker like Keds), shoes without straps to keep them from falling off your feet, and light colored clothing if you’re prone to sweating through your clothes.