Compatibility - personality type, mutual goals, similar outlooks on the game, and your instructor's ability to relate to your individual needs.
Accreditation and Experience - Does your golf instructor have the education and experience to take your game to any level you desire?
Many individuals claim to be golf instructors. Many of these individuals are self-proclaimed "experts," or had enough money to take a one to two week course on how to teach golf and make more money.
In seeking a golf professional to help you with your game, insure that the individual has an active accreditation with the PGA or LPGA, or, has demonstrated an inarguable ability to help players of many levels improve through many years of practice.
Video - Video swing analysis has become a staple for all golf instruction. It is no longer an option.
If you are paying for instruction that does not include video review of your game, you should reconsider why you are paying for golf instruction.
Video, when used correctly, is a third pair of eyes (you and your instructor are first and second). This is similar to a doctor using an X-Ray or M.R.I, to diagnosis a health issue.
Follow-up Communication - Your golf instructor should have a program available for you to utilize to ask questions, give feedback, receive follow-up instructions, schedule lessons, or make other inquiries after your lesson.