Wear something that you feel comfortable in that allows you to move around. There's no specific clothes you need for dancing, just stuff that's not too tight or restrictive.
You don't need dance shoes, trainers that are pretty slippy will do just fine. Otherwise if you want to get a pair of dance shoes, they're definitely worthwhile - we recommend SwayD, they have some great options. Just don't wear them outside, you'll ruin the suede soles.
If you do have dance shoes, we recommend you avoid shoes with a high heel. You're better off in fairly flat shoes for swing.