I've been grubbing out poison ivy from my property for the past two years, plus I have many decades of suffering a rash from it.
First, the "leaves three, let it be" slogan is stupid. There are so many ground plants with three leaves that this saying is meaningless. Simply train yourself to recognize it from pictures until you have confidence with identification. It's the waxy texture and serration of the leaves that help identify the plant. In winter it's the perpendicular hairs growing from the vines on the trees that identify it.
Second, the urushiol is like like axle grease. A simple "wash your hands" or rinsing will not be enough. You must scrub the skin as you would after working on cars. I use a brush and pay special attention to the thin skin where I usually get infected, like forearm, ankles, and between the fingers.