Is it the correct use of the adjective better when you hear or read "better educate" "better recognition" "better prepare"?
First, welcome to the forum!
To answer your question, all three examples can be correct. The only difference is that for "better educate" and "better prepare" the "better is being used as an adverb because it modifies a verb.
For "better recognition" the "better" is, as you suspected, an adjective because it modifies a noun. If you had said "better recognize" instead, than the "better" would be an adverb again.
Hope this clarifies things for you.
Welcome.
Of course, I agree with everything that Heimhenge writes above.
If I might speculate, reading between the lines, some people might criticize the verb phrases "to better educate," "to better prepare," "to better recognize," etc., because of the so-called "split infinitive" rule. This fake rule has been discussed a few times by Grant, Martha, and participants of this discussion board. The rule has no merit as a genuine rule of English grammar or style.
You can dare "to boldly go" and to freely "split infinitives" with assurance.
Glenn said:
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You can dare "to boldly go" and to freely "split infinitives" with assurance.
Except when your boss firmly holds to the split infinitive rule and is too closed minded to accept teaching on the subject. 
Yes. Also there are times when "boldly to go..." sounds better to my inner ear. But maybe my inner ear is just a snob; I'm not sure.