Well, tango is a sexy dance, so people tend to wear pretty sexy clothes, especially at the milongas (social dances), whereas the classes tend to be more casual. In either case, you will want to wear a blouse/shirt that allows you to easily raise your arms into an embrace. Wear skirts/dresses/pants that allow you to move freely and take large steps if necessary. For men, slacks and a button down shirt always work well. Many women choose to wear dresses or skirts for dancing, but that is personal preference. If you choose a form fitting pencil skirt or dress, make sure it has a slit so that you can still move freely.
The most important part is probably your shoes. If you don't have actual dance shoes, no worries ... for men, a leather-soled dress shoe is probably best (leather allows you to pivot more easily than soft rubber soles, for instance). Any shoe that allows you to pivot is key. For ladies, heels are recommended, preferably shoes with an ankle strap or T-Strap so that they stay on your foot well (as opposed to a pump, for instance). We sell Turquoise Tango Shoes at the studio (they are my favorite tango shoes of all time; I import them from Turkey), and if you get into tango, you will eventually want to invest in a good pair of tango shoes.