Since I'm still new to driving, I'm very conscious of the time I take getting my horse harnessed, and since I am chronologically challenged to begin with, I try in all areas of life to set myself up for good time management. It doesn't always work, hehe, but this is one area where it really does help...
Like Cotton, I also use the trick of setting out the harness on the fence prior to harnessing. Even if it's not in sequence, it still helps to see everything laid out.
This trick is a big DUHH that Leia pointed out to us: Mingus used to resist taking the bit, but it was because I wasn't undoing the chin strap. Once we did, it was easier for him to open his mouth to take the bit, and now he is fine with it and bridling is much, much easier.
I like to braid his forelock just to keep it from getting caught up in the browband and winker stays. I don't bother with a rubber band, as the braid will stay long enough for this purpose. I also like to fuss with his ears as little as possible, since the klutzier I am, the less patient he is, so I try to get it right the first time.
One of the biggest time wasters is questioning or rethinking things I already know -- like which hole to use for various buckles. If the buckle doesn't crease the strap, you might mark it to make it obvious, especially if you're using the same harness on different horses.
I keep one side of the shoulder hanger attached to the breast collar and bring it over the shoulder and neck. For now this works best for me, but I can see in time switching to Leia's method of placing the entire assembly over the head.
I place traces and driving lines safely out of the way and where I want them in the first place -- no undoing or rerunning traces under tugs, breeching, etc. I find it extremely annoying to have to redo things I should have had correct in the first place.
By far the biggest consumer of time for me is getting the holdback straps just right. as I'm still dinking (technical term, you know...) with the footman's loops and wrapping the straps around the shafts and how this affects the tightness of the breeching, etc.
This seems to differ every time we harness and hitch, which drives me nuts. I can see the wisdom of keeping the holdback straps on the shafts and then just attaching them to the breeching, but I haven't quite managed that yet.
Other than that, I think we get a bit speedier every time we harness, and Mingus gives me fewer dirty, impatient looks. The good news is that if he's impatient, it's because he really enjoys driving and wants to get going -- enough already!