Hiking generally refers to long nature walks taken from a base camp. You usually don't carry all of your gear with you, but it is a good idea to bring along water, snacks (extras for the kids!), a small first aid kit, a map or trail guide, and any clothing that might make a sudden change of weather more bearable. Be sure to keep to established marked trails when hiking with young children--this is not the time to go trail blazing or bush whacking!
When considering a hike with children, allow for more time than is generally estimated in field guides etc. as you may need to rest (and snack) more often, and also take more time to explore any snails or insects discovered along the way.
Before setting out, be sure that everyone is wearing appropriate footwear. If you are not seasoned barefoot hikers, then you will need at the minimum a decent pair of walking/running shoes for easier terrain, and light hiking boots for more challenging areas. This is not the time to be wearing open-toed sandals or crocs.

Canoe tripping is a way to travel as you camp. Canoeists camp at a different site each evening, packing up all their gear each day and paddle to a new destination to set up camp. Although canoe trippers may occasionally stay at one site for two consecutive nights, they usually move on each day following a loop, out-and-back or shuttled route travelling along rivers, lakes and ponds, and portaging in between bodies of water, over dams, and where there are rapids or waterfalls. Portaging means "carrying". When you portage on a canoe trip, you carry all your gear until you can re-pack your canoe and continue on. Canoes are usually carried by a single person by inverting the boat and wearing it "hat" style, balancing the centre thwart or carrying yoke across their shoulders. Two people can carry a canoe, but this is actually more difficult, and those people don't tend to remain friendly for very long.

Canoe tripping is a great way to get away from "the things of man", at least to a certain degree. It provides lots of opportunity for wildlife viewing as well. The scenery changes as you explore new areas each day. When canoe tripping, you can afford to take more gear than you could when backpacking, because you spend most of your travel time in a boat. Still, you will need to prioritize what you bring more than you would when car camping (see equipment for suggestions, and also food).

If you feel a little hesitant about canoe tripping, try just doing an overnight by paddling across a small lake to a campground for the night. In Algonquin Park, there are two "paddle-in" campsites designed especially with this purpose in mind.

When planning a route with children, consider paddling for 1-3 hours a day, then spend the rest of the time exploring the campsite, swimming, hiking, and otherwise playing around. Plan on an extra day in case you become wind-bound. If you are on a large lake and the wind gets strong, pull off the lake and wait it out. Often winds die down around sunset and again around dawn. Always err on the side of caution when making weather decisions on a canoe trip.

When you walk through the forest, you become a part of it. We often spend so much time moving quickly driving, boating, flying, etc. that we miss a great deal of detail. Walking changes all of that. You see things you would have otherwise missed.
Backpacking is just like hiking, except you carry all of your gear with you and camp in a new location each night. That means all your food, sleeping bag, tent, clothing, pots, stove, and anything else you bring along. Backpackers need to pack light! For this reason, they often carry dehydrated meals, and keep their equipment pared down to the bare essentials.
When backpacking with children, allow for much shorter days, and a slower pace. Carry lots of high-energy snacks and memorize many camp-style songs and games to keep their focus away from their feet and on more interesting things, like having fun. Remember that kids will not be able to carry much--aim for about 20% of their total body weight as a maximum.
Keep the first trips short--leaving them wanting more at the end of the trip. As much as possible, plan your trip to pass by areas that will be of interest to them, such as natural caves or waterfalls. Do some day hiking ahead of the trip so they get used to longer walks. Research the local flora and fauna, and local history including any myths or stories you think might appeal to them. Share some ahead of time, but keep a few to tell throughout the trip as well.
One book with some good basic suggestions is Camping and Backpacking with Children by Steven Boga.

By car camping, I mean any camping that is done where you drive to a campground and you use a motorized means to do your travelling. While it may not be a true wilderness experience, you may still have a chance to connect with nature by day hiking, exploring the local waterfront or joining guided interpretive programs.
If you car camp, you will likely stay at one site for the duration of the trip. This means that you will get to know the campground very well.
Since you will have nearly unlimited room to pack your belongings, you may choose to bring along home comforts, sporting equipment, etc. to help you occupy your time. Some campgrounds offer electrical and even sewage service for those who prefer to use trailers and RV's rather than camp in tents or under the stars.

While those comforts may work for some vacations, if you are looking to introduce your children to nature, you'd be best to leave those things at home and get back to the basics. Television has no place in the wilderness!
Carefully considering what to pack is a great way to open up a discussion with your kids about what people truly need to survive, versus what they desire. Consider the equipment list I've supplied as a starting point.