Trot, trot, trot. And lots of trotting helps. It helps both you and your horse.
Working up to 10 to 20 minutes cantering between long sets of trotting is good for their wind, and will get you very comfortable at that speed. Of course, you start and finish at the walk.
I generally start a soft horse (but has been at pasture) with 20 minutes walking, 10 minutes trotting, 20 minutes walking. Not a grazing along walk, but a forward walk.
After a week or two, depending on the horse, I'll do 20 walk, 20 trot, 20 walk.
A couple weeks of that, and it is something like 10 walk, 20 trot, 10 canter, 20 trot, 10 walk. They are getting more efficient at warming up and cooling down by then.
A maintenance program is about 10 walk, 20 trot, 20 canter, 10-20 trot, 10-20 walk. Then they are pretty fit.
I do this, with slight adjustments depending on the horse, for polo horses, horses that have grazed all winter but will be put to work for the spring/summer/fall on ranches, and horses that guys have hardly ridden and plan to take to the mountains elk hunting in the fall.
If I have a bunch, I pony two while riding a third. My youngest ponies four while riding a fifth. And we have friends that will pony six while riding a seventh.
I asked one of them, "What do you do if you have a wreck (a horse stops, leads get in the wrong place)?" They say it rarely happens (admittedly, I've not witnessed a problem), but at that point the advice is "Let go!" The horses either continue to follow the pony horse, or stop to graze, and the rider gets it all sorted out.
Have fun. Have you looked into Hunter Paces for starting out? Or going with the second field? I've not done a Hunter Pace, but they sound like fun.