Research the gallery you are submitting to. Make sure you’re not wasting your time submitting to galleries that would not show the kind of work you make.
Inquire about whether the gallery is actually accepting submissions. Your portfolio in the garbage doesn’t benefit anyone.
Respect the gallery staff, regardless of who you talk to. Being rude to the intern will not land you an appointment with the curator if the intern schedules appointments. Becoming friends with the staff is a good way to get a positive recommendation.
Attend openings or exhibitions that the gallery puts on. You are more likely to meet and talk to staff, who will then regard your submission inquiry with more care.
Contact other artists who show at the gallery. There’s no guarantee they will reply, but if you politely ask questions or for advice, this might be your best insight into the workings of the gallery. Many artists have websites and are easy to reach via email.
Be pushy or think that demanding attention will get you attention. (It is more likely to get your submission in the garbage).
Be unrealistic about your career. Submit to galleries who show artists in similar stages of their careers.
Request a critique of your work. That’s not the job of a gallery and you end up looking like an amateur.
Harass the staff if your submission has been declined. Instead, ask if you can keep them updated about your future work via email.
Lose hope. Persistence, hard work, and a good attitude will pay off.