Great article and cute photos of your family.
Your daughter is lucky to have you as parents who are getting her exposed and comfortable with the outdoors at a young age.
My daughter is now 6.5 yrs old, but we have been taking her out on short weekend trips (3 days the longest so far) since she was 2.5 yrs old.
After my daughter, Mei, turned 3 yrs old, it was a mix of having her walk with us and me carrying her on my shoulders with her grabbing hand fulls of my hair to hold on...or my ears. :)
In my case, my daughter is small/light (she is only 42 lbs now as a 6.5 yr old) it was not too hard to carry her on my shoulders while she was 22 lb to 30 lbs.
I thought of it as carrying a "normal" load of 40-50 lbs.
My experience matches much of what Tony Burnett has said....find a good place to camp at with a nice "pay off".
A lake is a great place to let play in the water...small boulders to climb on or around are favorites for my daughter.
Expect a pace of no more than 1 mile an hour with plenty of breaks.
The longest that my daughter has hiked has been 5.5 miles in one day while she was maybe 4 yrs old, but that was with a group of some eight other kids.
The more children around the same age, the better.
They keep each other motivated and "distracted" so that they will just fly down the trail.
Other things that I have found out that works for us is that my daughter has a small day pack from REI that has a 0.5 Liter hydration system.
I put in some powdered drink in there, which allows Mei to drink as she pleases and we don't have to stop to give her water.
Also, having her carry her own "treasure" mesh bag is a nice distraction.
She has picked up a few small, pretty rocks, flowers, and stuffed them into her treasure bag to look at later. This sometimes can be a nice distraction or game to play for her to look for something on the trail.
One of the things that she enjoyed the most was while in the Santa Cruz mountains, she was hunting for yellow banana slugs...thankfully, she was not putting them in her treasure bag.
As Mei has gotten older, I have done night time treasure hunts by taking chocolate "coins" with us to hide around the campsite for her to find.
Small glow sticks attached to a string are fun to at night for other children and her to play with.
Involve your daughter in some of the camp chores as she gets older...llike gathering fire wood.
My daughter loves to put wood in the fire.
Surprisingly, my daughter runs really warm...maybe most kids do. As a result, I have often only used my 40 F down sleeping bag for her to use, tucking the excess/unused part of the bag under her. She has burried herself inside of the bag and with her clothing on, she has been warm and toasty, even in chilly weather. (it might seem disturbing to have your child "suffocating" deep inside the bag, but my daugther has been fine).
Also, my daughter does not seem to care if she is sleeping on a thermarest pad or a foam pad...hell, even without a pad (as a result of tossing and turning she often is off the pad), so don't worry about needing to use a heavy, cushy pad for your child.
For safety, I have given my daugher her own headlight, which she keeps in her pack and I have a large orange safety whistle for her. We make her now carry her own rain gear, fleece jacket, hat, gloves, and a few snacks...nothing too heavy but everything she needs to be safe if separated from us for a short time.
She has been taught to always have it around her neck and if she is separated from us, she will stay where she is and to blow it three times and wait for us to use our whistles to respond back with two sharp blows. We have drilled this into her head quite a bit and had her practice a few times blowing it. (At first she was hesitant to blow on it for fear of being to loud for people around her....we have her carry this whistle when we go on vacations too even when we are in urban areas).
Anyway, just ranting here about what I can think of that might help you out.
Would love to hear what other parents have to pass on as advice.
P.S. The Rainshadow 2 Tarptent has been a great and recent addition for us as a family.