In many places (certainly all yoga & dance studios) outdoor shoes are forbidden but, even where they are not, we discourage their use. The floor will have been swept (and maybe mopped) so that dancers can roll or go on all fours. You’ll also want a greater connection to the ground than walking shoes provide.
1) You can dance barefoot.
2) All little more protection is given by “dance paws” (see http://dancepaws.com/)
3) More support & protection is given by dance shoes. Do get split-soled dance shoes (bendy in the middle). They will allow your feet to be active and connected to the ground. 3a) Split-soled jazz shoes are one up from dance paws;
3b) Split-soled sneakers give you more resilience but take you further from the floor.
A good local source for both is Les Gals in Silver Spring (http://www.lesgalsactivewear.com/).