What you said is like saying "I go store." It's grammatically incorrect (missing preposition to & article the). Similar thing.
I'm not going to explain what は、が、です mean. I don't have time and anyway that is covered in any beginning Japanese lessons. I found an online reference that could be helpful. Here is WA & GA and here is DA(DESU). But still you should follow the lessons in a good textbook. Good luck.
Adults omit particles all the time too, in casual speech. It's probably better, however, to use all the particles until you've heard enough casual speech to know where it's acceptable to omit them. I feel like in this case, わたし、おんがくがだいすきです is fine, but I'm not a native speaker so my 'feeling' could be wrong.
It's also good to pay attention to where わたし（は） can be omitted; in many case you don't need to specify yourself. It's often understood, especially in expressions of tastes, desires, and opinions, that the speaker is the unstated topic or subject.
But there is nothing unnatural about the way they were dropped in the original sentence. Nor was there anything inherently childlike in the way they were dropped. Dropping particles is one of the best ways to BS your way through not being sure which one to use and avoiding having anyone find out. It won't work for everything, of course, but the ability to speak in incomplete sentences is a feature of Japanese that can work in your favor.
I agree that she needs to learn them before she drops them.
Keep in mind the OP wasn't talking about spoken Japanese. In written form it's very unnatural to drop them. I'll go as far as to say it's outright wrong. As we all agreed, colloquially, it can be ok in the right context to drop the particles. In the wrong context it sounds uneducated or childlike -- young kids haven't picked up much grammar yet nor the rules of polite speaking. My five year old speaks like this.