「てくれた」implies "to me" or "for me". So it would be 田中さんが［初めて/最初に］教えてくれたアプリ

I don' think either sentence is correct. I can't understand what you mean by 初の. Do you mean, "This is the first application Mr. Tanaka introduced to me."? If so, you can say like this；これは田中さんが私におしえてくれた最初のアプリです。 By using おしえてくれた instead of おしえた, you can express thanks to Mr. Tanaka.