I'm probably the LAST one to give advice about this as it takes fussy me forever to get harnessed and hitched but I've found two things that speed me up incredibly.
1) Harnessing for show with French Tugs and no breeching: with the exception of getting that darn martingale on, even
I can get a horse harnessed and hitched in under five minutes!
I mean it's buckle the girth and crupper, pull the breastcollar on over the head, bridle the horse, attach traces to singletree and buckle one little overgirth strap and you're off!
I find that amazing.
2) When using my carriage harnesses, boy is it quicker with open tugs than with wrap straps! It reduces all that wrapping and snugging and buckling and double-checking down to one quick buckle (I leave the other side buckled all the time).
I guess the only other "trick" I use is leaving the neckstraps buckled on my breastcollars. I turn the collar upside down and pull it over the head just like a neck collar so all I have to do is right it and tuck the traces up out of the way. I can't imagine how much longer it would take me if I had to do up those double-buckles all the time.
One thing I will
never skimp on is taking the time to run my hand under the girth, breastcollar, crownpiece, etc. to make sure nothing is pinching or uncomfortable for the horse. I also stretch out his forelegs before we get going to free any trapped hair or skin.
When I unharness I leave the crupper buckled (lifting the saddle back towards the crupper then dragging the whole assembly down and off over the tail) so that it maintains its shape better in the harness bag.
Leia
Edited to add: The "Quick Hitch" stuff from harness racing is an assembly whereby the shafts actually click directly into an attachment on the harness saddle, no traces required. Very cool, but probably not a good idea for a little horse hauling more than its own weight!