In K-12 school ‘gym’ is a common name for a class. It’s also called ‘PE’ for ‘Physical Education’. With ‘gym’ used this way, various articles and prepositions can be used. For example: He’s generally a terrible student but he’s great in gym. My daughter hated gym. At gym, we had to go outside. I liked it when the weather was decent. In your example, ‘gym’ has a different meaning and needs an article: ‘the’, ‘a’, ‘my’ etc.

“So I started working at the gym as a trainer.” - this is correct. Without the “the”, this sentence would be incorrect. 🌈

"Working at gym" sounds odd, which leads many people to tell you it is wrong. However, "wrong" has many possible meanings: 1. not grammatical 2. grammatical but does not make sense 3. grammatical and makes sense, but we just don't say that. I vote for #3. Compare the words "gym" and "school". With "school" it is perfectly normal to say both "She works at school" "She works at the school". That is because we are comfortable using "school" as the name for a class of institutions. We don't we feel comfortable using "gym" that way? I don't know why. We just don't. Maybe someday we will change and use "gym" that way. There is no grammatical rule preventing us from doing that. But until then, just say "at the gym".

At the gym, <something happened>. I was at the gym when <describe event>. Being at the gym was <describe feeling>. While at the gym, <describe event>. Today, at the gym, I <describe situation>. He left his wallet at the gym. (used to describe a location) The cost of food at the gym is expensive. (used to say where something happened). There are many situations in English, where we use "at the <place>". If you take away "at the" and you're left with the word "gym", there are more ways to use it as a noun. The gym was cold and dark. I was at gym class when the lights went out. I lost my phone, but it was in the gym. When a speaker omits "the" before gym it is implied by context. "Each day at gym, I disliked running." We know the speaker is in the gymnasium. We know the speaker is there for some type of class, lesson, or practice. We know the speaker is using the word gym to mean a location.