Ever since I moved to Elizabeth, I've been using Jones Dry Cleaning. Dry cleaners are generally one of those kind of places that you only hear about when they screw something up, so I want to toss some praise their way.
All the staff are very friendly and efficient in ringing up a ticket and getting your garments off the rack. They offer a discount (20% I believe) if you drop your things off on Wednesdays. They also offer same-day cleaning.
I had an issue one time when I received a dress shirt back and a button was missing. I took the shirt back the next day to ask about it and they immediately apologized, took the shirt back, and had it ready for pick up the next day with a new button.
Great customer service and a great experience all around.