You are right that the Arduino is a good place for you to start learning about electronics and programming. The children are quite young though and to begin with I would start them with something even simpler than mssltd's suggestion. Just wiring up a battery some bulbs, push buttons, switches and buzzers (i.e. electric rather than electronic) will get them into wiring and logic. I remember finding electromagnetic bells fascinating.
Since you are good with your hands, gears, pulleys, motors and microswitches can add to the fun and you will be able to introduce your electronics and microprocessor skills when they are a bit older.