Ok, now I am obsessing. ‘Since 3 years ago’ at the end of this sentence, jars. You can use this phrase, but it works best as a fronted adverbial; a phrase at the beginning of a sentence that describes the action that follows. However, more natural is to use ‘since’ to describe the point in time that the action started, defined from that point: i.e I haven’t been kissed since last summer or Since last summer, I haven’t been kissed. ‘Ago’ defines the point in time from now: i.e two years ago, I broke my leg or I broke my leg two years ago. So ‘Since 3 years ago’ jumps back and forth. However, this can be used in certain circumstances to describe the period from then until now, it’s best to get it over with at the beginning of a sentence. Of course, it can be replaces with ‘for 3 years’. Hey, that works better: I have been learning English for about 6 years but more seriously for 3 years. Or to clarify it was the last 3 years of serious study and it wasn’t the middle 3 (although that is inferred), how about I have been learning English for about 6 years but more seriously for the last 3 years.

Great! You may also use however instead of but and go into new sentence. I have been learning English for about 6 years. However, I have been learning English more seriously since three years ago.

Firstly, punctuation helps to show the natural pauses separating the general study time from the more intense. It would be more natural to say: ‘I have been learning English for about 6 years but in the last 3 years, my study has become more serious.’ It could be emphasised by adding: ‘I have been learning English for about 6 years now but in the last 3 years, my study has become more serious’. You could soften the sentence, depending on what you are trying to stress and make it flow more naturally. For example: ‘Although I have been learning English for about 6 years, in the last 3 years, my study has become more serious’. I believe the original sentence doesn’t work because ‘but’ is a conjunction; joining two sentences together. The first; I have been learning English for about 6 years, works well. The second; more seriously since 3 years ago, doesn’t. It would be preferable to say ‘since 2013’ or ‘3 years ago, I started learning more seriously’ Does this help? I’m not a teacher; just a native speaker.

It's meaningful sentence