Wear what is comfortable to you. Some of the students wear sneakers, no shoes just socks, flats or traditional Chinese shoes. We practice in a large room with a wood floor. The floor is a bit tacky with rubber soled shoes and slippery if you are in your socks.I recommend trying a variety of shoes throughout your training. Using different footwear will help you become comfortable practicing Tai Chi on surfaces with varying levels of friction.
Here are the shoes some of use wear to TaiChi class. These have a rubber sole so you can wear them outside. The sizing is fairly accurate. I wear a 10.5 shoe and I found the 46 fits me well. The 45 were a bit too small. I have heard from some of my students that they have found the Chinese shoes in US sizes on amazon so feel free to look around.
Find them here on Amazon.