Boxing apparel doesn't have to be elaborate. I used to have a membership in a women's health club years ago. A vendor came in with racks of fancy workout clothes one day while I was there. There was a shopping frenzy among the women, and I admit, I got caught up in it, too. But having designer workout wear doesn't make a bit of difference. Wear what you feel comfortable in.
For regular visits down to the boxing gym, a T-shirt, shorts or workout pants, socks, a sports bra, and a pair of athletic shoes is good enough.
Later on, you may want to buy a pair of boxing shoes, especially if competing is a goal. Boxing shoes should be lightweight because boxers have to be able to move fast around the ring. They also should have good traction to cut down on slipping, sliding, and falling.
If competing, boxers need to wear jerseys, sports bras, and boxing shorts in addition to the boxing shoes. In many instances, officials will not allow people to compete if they're not dressed properly, especially at sanctioned amateur matches. Substitutions can be made, however. The jerseys and the shorts don't have to have the word "boxing" written on them, for example. Athletic shorts may be worn instead of regular boxing shorts.