For a beginner class, special dance shoes are not necessary.Wear shoes with leather or other hard soles that are somewhat slippery and will not catch on smooth floors. You want to be able to spin and slide... not stick. For men, a thinner, more flexible sole is better than a thicker, rubbery sole. For women, flats or heels are fine, ideally shoes with support around the ankle. Avoid shoes with an open back (i.e. flip-flops or mules).
At a more advanced levels (Improver, Intermediate, Advanced, or Master Classes) especially for followers who spin more, shoes with soles that are conducive to spinning will be required (thin leather or suede soles are ideal). Running shoes are not recommended. However it will be a while before your normal shoes present a problem. It’s probably best if you wear the same shoes you wear when you go dancing. For example if you wear 2-inch heels when you dance in the clubs, you should also dance on 2-inch heels to get accustomed to the different balance involved.
If you take dancing more seriously or for a long period of time, you may want to invest in a pair of dance shoes (thin suede soles) which cannot be worn in the streets and a pair of dance sneakers (jazz shoes) to practice your shines or for a full day of workshops.