My favorite advice for new campers is to leave the axe/hatchet/machete at home and bring a hammer (I prefer what is called a drillers hammer - short handle w/heavy wide head) and a bow saw or camp saw. Be sure to get a blade cover for the saw, or make one from a length of radiator hose.
You will drive tent stakes more easily with a wide face hammer, and cut ALOT more wood with a saw than you will with an axe or hatchet and it is much safer to use.
If you want to take a blade, an EDC pocket knife is a great choice. If you let the (older) kids carry a pocket knife go inexpensive to begin with because they WILL loose them the first day. Get a blade that locks while in use for folding knives. When camping with kids, show them how to use the tools you choose to bring rather than banning them from helping.
The last essential in my camping kit is always a Frisbee or Aerobee throwing toy. Not only fun to play with, but they make great paper plate holders if you get the right size. Best place to carry them is in the end of sleeping bag stuff bags. They do not get bent or crushed as easily, and they help keep dirt and rain out of the stuff bag.
This man knows of which he speaks. Secrets of the trade you could say. It took forever to convince the scouts that a saw was better than a hatchet, but it certainly is. (but I love hatchets!)