You can do either.
You can get your costume pieces and change at home; or you can keep them at work and change there (depending on locker availability). You can turn them each week for cleaning and get new ones; or you can wash them yourself. Or you can do any combination of the above, it's your choice.
Until a few years ago, Cast Members had to get a new costume each shift and change at work. As such, those Cast Members who have been with the company for a while (as well as those who also work at or have worked for Universal Orlando, which still requires this) will do that. Those hired more recently will generally keep their costume pieces at home, change there, and wash their costumes each week for the convenience factor. However, it's a choice left up to individual Cast Members.