They're both correct; they just have subtly different meanings.
In "I will learn better English," "better" is an adjective modifying "English."
In "I will learn English better," "better" is an adverb modifying "learn."
So in the first one, you're saying you will improve your English. Yes, as you say, English is English, but it's common when talking about acquired languages to talk about the quality of your language to mean your fluency and facility with it. So, "your English is good, but my French is terrible," is completely acceptable, and therefore, so is "I will learn better English." You're saying simply that you will improve your English skills.
In the second one, you're saying you will learn better, which has more of an implication of working harder and being a better student.
In terms of basic meaning, it's much the same, but the difference is tonal: in the first version, you're talking about picking up skills, but in the second, you're talking about committing yourself to them.
And to your last question, about a "better" way to say it: to me, a "better" way to say something means that it better expresses your meaning, so it's not for me or anyone else to say what's better for you, but if you're looking for alternatives, some possibilities might be:
I will improve my English.
I will hone my English skills.
I will keep practicing English until I get better.
I will learn conversational [or fluent] English.
...and so on.