I prefer dogs better than cats
I like dogs better than cats.
Are any of the above sentences correct?
Both are incorrect. As a rule "prefer" should be followed by "to" or "over".
I prefer dogs to cats
or
I prefer dogs over cats
I think it'd be more natural to say:
i. I prefer dogs over cats
ii. I like dogs more than I like cats.
Omitting "I like" in the second sentence will cause ambiguity, but it's still grammatical.