Sulu's first command is the Excelsior which is a pretty important vessel by Starfleets standards
Do we know for a fact that it is his first command? All we know from TUC is that he commanded her for three years at that point. There is a sizable gap between TFF and TUC (I think about 9 years, but I think its about 6 in the Okuda Chronology), which leaves a few years for a first command, perhaps a destroyer class vessel as a commander.
"Canon is only important to certain people because they have to cling to their knowledge of the minutiae. Open your mind! Be a Star Trek fan and open your mind and say, 'Where does Star Trek want to take me now'." - Leonard Nimoy