Practicing tai chi at home, loose "lounge wear" is quite suitable, though you probably wouldn't want to go out in it. (Not necessarily. Some of my students turn up for class looking like they have just rolled out of bed!) Sweat pants are also fine.
Martial arts pants and tops are quite ideal. Tai chi/king fu uniforms are readily available and start at about $20. The tops are chinese style tops, "frog buttons" and all, and the pants often have elastic ankles. The good thing about the pants is they have a deep gusset at the crotch. That makes for a lot of freedom when bending, stretching etc. Karate/aikido/judo pants perform the same function. Cotton or poly/cotton are very serviceable and throw in the washer. If you want to look like the folk on the tai chi videos you will have to buy a silk suit and it will cost you more like $80+. (Dry clean only!)
Shoes? I read that the shaolin monks, for training, wear a simple cotton and rubber "Feiyue" shoe. I found some for $15 and bought them and used them for a while. They are very light and quite comfortable. My problem with them is that the sole is rounded rather than flat. This makes them a little unstable. I know that sounds silly because they have barely any sole at all. That is just how it feels to me. After wearing them for several weeks of daily practice, I tried my Converse "Chucks." They seem just right for tai chi. The sole is perfectly flat and, for me, they provide more stability than the Feiyue. Either shoes are vegan friendly, which is a good thing.
I have since realized that the $15 Feiyue shoes sold by many martial arts stores in the US are fakes. You get the real deal for 50 euros from Feiyue shoes web site. Well I never!