The sequence of procedures used in tacking and gybing isn't set in stone. It depends to a large extent on how the boat is rigged and the type of boat. You have a limited amount of time to accomplish all the tasks necessary to tack the boat, so, your challenge is to figure out how to do all those things in the time available.
I suggest you begin to improve your procedures by analyzing each separate step of the whole process. Look for procedures that can be done either before you put the helm over, or after the boat is on the new course. For example, sometimes you can either leave the mainsheet traveler centered, so that you don't have to readjust it each time you tack. By doing so, the mainsail becomes self-tending. If you're racing and want to readjust the traveler, you can often make the new adjustment before you put the helm over. It's easier to readjust the traveler before the tack, (by easing it to leeward) than it is to pull the traveler up to windward after the tack.
On a keel boat, you can usually tail a jibsheet much faster if you're standing, rather than seated, because you have more room to swing your arms. If you reach out in front of you and pull the jibsheet as far behind you as possible on each stroke, you can pull in the jibsheet much faster, because you're tailing nearly twice as much line with each stroke.