The word kit in the sense of clothing/equipment is BE, as far as I know. Gear in this sense is also probably more BE than AE.
Gym and P.E., on the other hand, are fine in AE, but they refer specifically to the required class in school, not to other sports activities.
Sporting goods and to some extent sports equipment would be more for things in stores, or at least in larger quantity.
Attire is a more formal word, used more in writing; in normal conversation we just say clothes.
For the sentence in the original post, 'Ich bringe mein Sportzeug mit,' things for sports, stuff for sports, maybe just sports bag (assumed to contain whatever it is), anything like that would be fine.
In practice the sports part might be clear already from context, so you wouldn't really have to use the word 'sports.'
A: How about a game of tennis tomorrow after work?
B: Sounds great. I'll bring my stuff.
A: Mom, don't forget I have an extra soccer practice this afternoon.
B: Well, don't you forget your stuff.