Clothing for Comfort
No you don't need special clothes or shoes to enjoy dancing. This isn't a fashion show. But you will have problems if you wear something that restricts your movements, or that makes you feel uncomfortable. So here's a few suggestions to help you and your partner be more comfortable on the dance floor:
Wear shoes that stay on your feet
Open-back shoes are a bad idea. If you can't go forward, backwards, sideways, and turn without coming out of your shoes, don't wear them dancing.
Wear shoes that let you pivot
Most athletic shoes and many synthetic-soled shoes are too sticky for comfortable dancing. You should be able to pivot easily, and be able to drag or slide your foot without it sticking to the floor.
Don't wear big shoes
Sounds dumb, but it's true. Big clunky shoes, or shoes that make your feet look big, will make your feet feel big and clumsy.
So what do I wear?
Short answer: Experiment, and wear whatever works for you.
Long answer: Most people find that leather-soled shoes work well on a wooden floor. Heel size doesn't matter--the important thing is to feel comfortable and stable. A flexible sole works better than a thick, stiff sole. If they're too slippery, try using a wire brush on the soles. Real ballroom dance shoes have suede leather soles. Jazz flats have the same soles but usually cost less.
Wear clothes that let you move
Guys are usually okay with this, but women sometimes show up to dance in straight skirts or tops that fall off their shoulders. If you can't comfortably take a full stride in every direction, or raise your arm above your head, wear something else.
Wear clothes that are comfortable in a class setting
You're going to be in close contact with people you don't know, which may raise some issues of modesty and comfort levels. Wrapping your arms around strangers can feel awkward, and there's no point in making it worse by wearing something that makes your feel exposed.
Shorts/short skirts: Are popular and easy to dance in, but remember that thigh contact is a normal and necessary part of some dances. If you're not comfortable having someone else's leg between yours, you might want something longer.
Low-back/backless tops: Only if you don't mind other people's hands on your bare back. Having to put a hand on your bare back will also make some partners feel awkward.
Big jewelry: Big earrings will whack you in the face, and sharp rings can cut or catch on things. Keep it simple.
Tank tops/sleeveless shirts: You might not mind strange hands on your bare shoulders, but some of your partners won't be comfortable doing that. The male armpit is also not generally considered to be a thing of beauty to be displayed at close range.
But there are no rules. Wear whatever you enjoy dancing in and find comfortable for class.