First of all, welcome to the forum! I'm glad you posted.
I think everyone approaches warming up a little differently and most of us even do it differently with different horses. My gelding for instance has no problems focusing on his work when we start but it takes him a long time to get supple enough to really get his hindquarters under him and power forward without overflexing in the neck. For him we do a lot of bending and flexing exercises at the walk then add some simple circles, figure eights and serpentines at an easy working trot until he's rhythmic. Once he's moving freely then we add shortening and lengthenings at each gait and increasingly frequent transitions until he's really working well. I know he's warmed up when he's taking up very strong contact, responding briskly and easily to all commands and feels collected and energetic in my hands. For this horse with his physical problems (he had locking stifles and is now recovering from a back injury) it's a lot of work to push from his back end so we take it slow and easy and do a lot of stretching in-hand before and after his workouts to help keep him sound.
I know friends whose horses are very powerful and energetic and their worst challenge is working the horse
down at the start of a session.
The horse doesn't want to focus and is raring to go so they have to spend almost an hour doing long trots and exercises designed to get the horse paying attention and thinking about his or her work. For these horses sometimes the best remedy is sweat!
Needless to say these horses are usually fit and have reached an advanced stage of training.
For a green horse I take things slow and spend a lot of time focusing on "whoa, stand" and good clean transitions. The horse must have a quality walk with lots of swing to it, a steady forward trot and should neither leap up into the next gait nor fall out of the last one when brought down. By the time they can give me these things they're pretty much warmed up as it takes a baby awhile to focus that much.
The only thing I do differently at a horse show is I may take the horse out on a lunge first to let him blow off some energy since he's been stalled. He can run and buck and stomp and do whatever he wants until I see him start to settle down, then I'll start asking more and more firmly for specific transitions and such until he's obeying well. That's pretty much the warmup and when I hitch I'll go straight to serious mental work as his body is already loosened up. I do make sure the horse has a chance to get used to all the other carts in the warmup ring and make allowances for distractions at first but eventually I expect the same performance he'd give me at home within the limits of the space available. It seems like we never get as good a warmup at a show as we would elsewhere because they're always switching directions before you're ready (or worse, won't switch directions at all!) and you're constantly stopping for a horse in front of you or getting run over by one behind. Yuck! I have a tendency to find a quiet grassy place in the parking lot and warm up by myself.
Leia