First off, "Well done Derek for introducing your kids to hiking"
I am a father of 3 daughters and have backpacked with them since they were around 6. Day hikes/car camping - even younger.
Here is some friendly/fatherly advice for you while hiking with children.
1. Take your time while hiking. There is no race here. Remember to "stop and smell the roses". With kids, stop and find magic wand sticks along the way (it is a fun game at the camp-site). Let them poke a bug or two or see who can toss a rock into the lake/stream and hit a large piece of wood/boulder. Make activities into a game if possible. *Remember, if your kids do not have a fun time with you on the first few outings, getting them to go again will prove to be more difficult in the future.
2. Regarding kids carrying something: I always let me daughters bring one small toy and they had to carry it in their own backpack. I also had them carry their own sleeping bag as well. Bags are usually light and take room - so it allows them to feel as if they are contributing to the carrying load and does not tire them out too quickly. As they age, they carried more.
3. Stop and force them to drink water and snack often. They will forget if you do not. Here again, hate to see a child not have a good time and if they run out of fuel, get tired and cranky... well that leads to not having a good time.... you get the point. Bring smores makings as well. That is always a high-light at the end of the evening. Marshmellows at a minimum. Just find a stick, carve it into a point and "wa-la" you are off and roasting. Besides, there is something comforting about a fire at night and smores add to the fun.
4. If you elect to camp out over-night, keep the kids warm. We have good 30' bags from REI (Which if you are in need of sleeping bags, let me know and I will sell them cheap to you as my daughters have all but out-grown theirs). The girls slept in sweats/long underwear. Hats if needed. I bought them all tiny Photon Lights to keep inside their bags for night. They could click them off and on and feel a little more at ease durning the dark nights. I also took a candle light and hung that from the tent. Nice ambiance lighting. Bring a deck of cards for games if the weather is poor. Beyond that, there will be more than enough "outdoor" activities to keep them occupied. We also have a book on "what if" questions or "if you had to choose, which would you rather...."
5. On the topic of medical needs: I know I took too much stuff in this category. Band-aids, anticeptic ointments, allergy meds (topical and oral medicines), blister needs, etc. There is not much weight here, however, if the need arises....better to have it than not. I always brought bug juice (put it on your hand, then apply to their faces (lightly) and I would spray liberally on their clothes. Again, back to the "if they have a good time (no bites) they will want to come back" and DEET washes out of clothing. I would bring 30-50 spf for the sunscreen and definately sweat-proof! I would bring their tooth-brushes and tooth-paste. We take sponge baths. Getting girls into a cold river does not happen... so not much need for soap. Bring the hand-sanitizer for before meals and after "duty calls". I bring rolls of toilet paper (think 4 women (Mom included)). Toilet paper is good. That is one item you do not want to run out of with women/girls. Geezzz....
6. Snacks: gummy bears (they do not melt) - I would take my girls to the store and let them create their own version of GORP (chocolate, nuts, etc.). We bring granola (mixed with strawberries, blueberries, nuts, etc.) for breakfast and lots of hot-chocolate.
7. Rain gear, we all have rain jackets and rain pants. However, we live in WA and those items are pretty much a mandatory for backpacking in this state.
8. Regarding daily milage... that depends upon the kids. I have had some days wehre my 6 year old would like 6-7 miles (half of which was in snow) and other days where we only made it 3-4 miles in great weather. The trip intenary HAS to be open - unless you are needing to get back due to weather or out of food. Again, remember to make this fun for the kids! If you do not make it to the ideal camp spot or locate - who cares. Next time you might be able to. When we would backpack for 3-4 days, then ideally, we would hike hard for the first day, find a spot as a base camp and then take day hikes from there. Kids like to know the plan (how far and to where you are going). For instance, a daughter would carry the GPS unit. A GPS lets them know where they are, how much further, etc. I (Papa Sherpa) would carry the map and compass.
9. I have a 6-person GoLite Shangra La with a bug net. I love that tent. Light and huge. Bought it years ago and at the time was thinking "...am I really going to spend this much money on a tent?" Needless to say, I bought it and been happy since. Remember, you have to carry what they do not... again, just call me "Papa Sherpa".
We have a tradition, perhaps you might like to incorporate as well: We go to a DQ after the hike and they can order any ice-cream thing on the menu. Again, makes it fun for the kids.
Well, I could go on and on. If you want more specifics or input on a topic, just reply back or send me a PM.