Somehow, I think "better English" is incorrect, because I think there isn't better English; English is English. But I hear this phrase from other ESL students a lot. Is this correct way of saying it? Or should I say "I will learn English better"? Or if there's any other better way to say it, I would love to know. Thank you.
It depends on what you're trying to say.
In learn English better, better is an adverb that modifies learn -- it means you'll improve your learning process.
In learn better English, better is an adjective that modifies English -- it means that you will learn a more proper version of English. For instance, In England, public school English or King's English might be considered "better English"; in America, the English used in major national newspapers or network news broadcasts is typically considered the standard for quality.
'I will learn better English' is standard. It is like saying 'My driving is erratic'. In that case, 'my driving' refers to the act of me driving, or to the manner in which I drive. When I say, 'My driving is erratic,' I mean that my acts of driving are erratic. 
In the case of 'better English', 'English' refers to my use of English, and 'learning better English' means improving my usage of English. This is gramatically the same thing as saying 'I will develop better usage of English'.
You could say 'I will make my English better'. 'Make' can take two direct objects, 'my English' and 'better'. In this usage, I make [the first object] become [the second object], so I make [my English] become [better]. However, there are very few words which do this, and most words only take one direct object. 'Learn' only takes one direct object, in this case [my English], so 'better' can't be another direct object. Since 'better' can also be an adverb, it reverts to acting as an adverb modifying the main verb, 'learn'. That means that if you say 'I will learn English better,' you are describing how you will learn, not what you will improve (for example, you will learn English more quickly, such as taking two years to learn it instead of five).
 In this context, 'driving' is technically called a gerund, but most people would think of it as a noun.  https://www.englishforums.com/English/VerbsWith2DirectObjects/plljx/post.htm