My daughters are 7 and 9. I have been taking them hiking since they were 6 and 4 in the White Mountains. I have never carried them an inch. This year we started tackling come of the 4k peaks.
First thing I taught them was to follow a blazed trail and don't wander away from adults.
Next thing was the essentials of the hug a tree program. Basically stop, think, observe, plan Stay where you are unless you can see a better location immediately. Make shelter, Make signal, wait to be found. We rehearse this information more than twice on every hike.
At 6 each got a camp-knife (chaep knife / fork / spoon hobo knife) and some instruction on proper handling (carry point down, don't put it down open, don't let go until reciever acknowledges reciept when passing) We rehearsed this at home with every handling of a knife. Made sure that they had opportunity to use their camp knives at every hike ( pealing oranges, cutting apples, preparing kindling)
At 9 (one year ahead of plan because she is more mature than expected) I am teaching the elder to make fires with matches. This is slow going because she is very tentative with the match (burned her fingers a few times already)
For every hike each carried their own gear and water in a backpack (yes even the 4 year old - those were short hikes < 2mi) Minimum gear was (and still is) space blanket, emergency poncho, food bar, flashlight, whistle, compass, warm sweater / windbreaker / coat - depending upon whether and altitude, 1 liter water, GORP (They like the kind with chocolate and mini-marshmallows!), FAK, TP, Towel. Most of the required gear fits in their belly pack - only the clothes and towel go in the backpack. Water is usually carried in hand because I haven't sprung for hydration packs yet for them. We rehearse the reasons for the equipment every outing. Daddy - "What do you do if it starts raining unexpectedly?" Kids - "Put on your Poncho!" Daddy - "What do you do first if you discover you are off the trail and lost?" Kids - "Stop!" etc. I got them used to using the compass to tell which way we were going for the first few years this last year I have taught them how to maintain a straight line travel by sighting ahead to landmark and moving from one to the next in steady direction, and that if you leave the trail in one direction you can get back by reversing. The 9 yearold is catching it but the 7 yr old is not completely getting it.
They have a lot of fun with Dad in the woods but they would never run off through the woods alone at this point. The most important thing I have taught them is that the woods are large and it is easy to get lost so don't.
I have to say I'm proud of my kids but was astonished a few weeks ago when we hooked up with another family with a 9 yr old. They are a bit more extreme than I and had already taught their daughter some technical climbing and that kid was a mountain goat! We all had trouble keeping up with her!