Is this sentence right?
Soon after this experience I really realize that the music business is the sector that I strive for in my professional career.

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I would agree with Sarah, leave out the word “really”, and I would also change “realise” to the past tense “realised”, as it is an event that has already happened. :)

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It is fine assuming you already explained the experience, but "really realize" sounds awkward.  I would suggest just leaving out the word "really."

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It's grammatically correct, however it sounds a bit awkward to a slight extent, i would personally say:

"soon after that experience, I've realized that business in the music industry, is what i should strive for."

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