Both phrases are correct. Depends in which context you are using them, easiest way to explain is by example. 

1. I like chocolate cake a lot better than orange cake. 
2. The weather is much better today. 

is "a lot better" or "much better" correct?

Both are correct. It just depends on the context.

You may also use the following:

Tons better
Heaps better

Both are correct. It will just depend on the context  or the situation where they are being used. 

They have both grammatically correct but there are some new answers when you do speak about those two grammatical structures.
A lot usually use as the middle meaning. 
And the mach when you describe something with relief especially with the word better 🙏
Of course there's a big pack of rules to use correctly a lot of and much, but that takes the whole lesson to the honest 😅🙏

Both correct but depending where and when u want to use it

Both are correct. 
Both are used in different context. 

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I study English language and literatureThe key to success in this kind of environment, Tanguay said, is the ability to collaborate. "You really need to be able to fill various roles in order to collaborate effectively," he said. "If you already have someone on your team who is going to be the one to critique all of the suggestions made, then you don't need to join in on that. Instead, maybe you need to be the person who is going to come up with creative ideas. You need to have that flexibility."

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is "a lot better" or "much better" correct?

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