During the Revolutionary War the Continental Congress issued a specification for a standard officer's uniform for the Continental Navy. I'd have to look up the specific details (such as rank differentiation), but the basic "color scheme" was a dark blue jacket with red turnbacks, a red waistcoat, and red breeches.
There was no standard uniform for enlisted men in the Continental Navy - or the British Royal Navy at that time. There's scarecely any pictorial evidence about what American sailors of the Revolutionary War looked like, but it's a safe bet that they were a pretty scruffy-looking lot - as were their British counterparts most of the time.
The word "Constitution" in the subject line has me a little mixed up. The U.S.S. Constitution was, of course, launched in 1797 - fourteen years after the official end of the American Revolution. The Continental Navy (and its uniform regulations) had long since ceased to exist. The newly created U.S. Navy had its own officers' uniform regulations (the details of which, again, I'd have to look up); most obviously the red breeches and waistcoats of the Continental Navy had been replaced by white. As was the case in the British Navy, there continued to be no official enlisted men's uniform.
During both periods, individual captains, if they could afford it, sometimes bought standardized shirts and pants for their crews to wear on dressy occasions - or for such "special details" as boats' crews. Again, the contemporary pictorial evidence is too scanty to help much, but it looks like enlisted men dressed in whatever they had with them when they enlisted - or whatever they could find in the ship's "slop chest."
I've seen pictures like the one lolok mentioned, but I have trouble believing them. My inclination, given the realities of shipboard life at the time, is to think that American sailors of both the Revolution and the War of 1812 looked about like their British counterparts - i.e., about like the guys in the movie "Master and Commander."
That's the best I can do without digging into some books. There probably are some uniform experts out there who can help more than I can. I think, though, that the most conspicuous difference between the Revolutionary and Early National periods would be in the officers' uniforms.