Saya ada di dapur. Would it also be correct to omit "ada"? Saya di dapur
Yes, in most sentences "ada" can be omitted. "Saya di dapur" would be a perfectly acceptable thing to say.
You do not need 'ada' but it's inclusion helps you learn new words. It is likely that a fluent speaker would omit it, but knowing the word means you would underrstand it when you ask for something and the reply is, ada. E.g. Andi di dini? Ada. Punya teh es? Ada. Ada is sometime used instead of Iya or Ya.