Not much to add to whats been said above.
I wouldn't worry to much about neck reining at the trainers. A few trail rides and the horse will learn it on the job. As you head down a trail, the horse can see the trail, and basically he knows where he needs to go. But you keep signalling him with neck rein queues. He will figure it out. I've never had to conciously teach a horse to neck rein. Just be an active rider while riding the trail and the horses will pick up your slightest queue.
I hate a horse that won't stand still, If I spot an elk or deer and want to glass it with some binos, I don't want my horse fidgiting around. I want him to STAND STILL.
I can't tell you how many times, I've pulled a rain coat or jacket off my strings and put it on while still in the saddle. I also take the jackets off and tie them back on the strings from the saddle. Make sure you can pull out a coat, shake out the folds and put it on with out the horses worrying.
I don't like a horse banging my knees into trees. It takes a while for them to learn how wide a berth they need to maintain when they go past trees. Probably not something your trainer will teach, But you need to teach. Also they need to learn to look up and see branches that would brush or knock me off the saddle. That is more difficult to teach. I've got a 15 year old gleding I like to ride. I can head up the canyon in pitch darkness and know he will skirt wide enough that I won't get slapped in the face with a branch in the dark. My younger horses all cut the corners trying to save steps, and I get the surprise of banging a knee or scratches on my cheeks. Now if I could just teach him to spot the spider webs!
My horses need to Hobble and Highline. I aways carry hobbles on trail rides. When I stop for lunch, hobble and let the horses graze. Overnight camps require highlines. The horses need to be comfortable being tied all night. They need to be able to eat all kinds of feed. At the truck I feed baled hay, at lunch they get to graze on grass, and if we packed in ( especially during late fall and winter when there is no graze) they get pellets/cubes that I have packed for them.
Spook in place, My horses will run into something along a trail that they have never seen. Something will jump out of the bushes and surprise them, I don't want to get slapped in the face with branches for the next hundred yards trying to get a spooked horse under control. Grouse, Pheasant, Turkeys all flush and fly off from under foot. Deer and moose jump up out of their beds, and bounce off. We ride past cattle on grazing permits, and it always seems the calf will be on the other side from the cow and they will sprint in front of us trying to get back. Add to the wild life, Hikers that don't talk with funny shaped back packs, mountain bikers, ATVs with loud engines, Dogs, strange animals like Llamas or camels. Even the surpise noise of lightning clap or a microburst of wind knocking over a nearby tree. They can spook, that's OK, but we are not going to allow them to spook and run.
Horses need to stand while you mount. No moving off until I'm in the saddle and signal my intent. I've been hurt too many times in the back country. And somes times it just hard to climb into the saddle with a broken arm, wrist, bundled up with heavy winter cloths, if you are a small person on a tall horse etc. If the horses is moving while you are trying to get on, It can be very hard and dangerous. They need to stand, no matter how hard I struggle to climb up.