Yes, they can play games - there are hundreds if not thousands of games available in the iTunes Store. Beware of the "free" games; many are free for the base game but charge for things like additional levels, "coins" etc. and those paid add-ons can add up quickly, to dozens or even in some cases hundreds of dollars. I'd strongly suggest you set up restrictions on the iPad to prevent your children from making unauthorized purchases. See:
You will need an Internet provider of some sort; Apple is not a provider and so the iPad does not include a data service. You can either use a home Internet connection with a WiFi access point (aka "router") or you can get an iPad with WiFi+Cellular and sign up for a data plan with a cell carrier. WiFi through a home Internet service will probably be cheaper.