Basic first aid is maybe a misleading. There is nothing basic about first aid, it's really complex and delicate, because it's based on the complex and and delicate ways that the human body works, both children and adults. Even though the principles are pretty complex, the procedures that you can carry out are very simple, once you get to know them. Knowing these simple procedures could save the life of a person in your family. Procedures that everyone should know (really, they should be taught in schools, but they're not) include how to tie a tourniquet, how to give CPR, how to sterilise a wound, and how to remove a splinter to prevent it becoming infected. If you have anyone in your family or workplace who is diabetic, epileptic, or has any allergies, then you should also be familiar with the procedures necessary to administer these conditions. Of course, you can't carry out first aid without a first aid kit that is up to date and properly stocked, so if you don't have one of these you are asking for trouble. You can buy kits from basically any first aid training company, the same places that put on first aid education courses.
It is also wise to keep the medical histories of the members of your family and the number of your family doctor handy in the first aid kit, in case of emergency. That way if there's a babysitter or a relative in charge, they'll feel equipped to deal with disaster.