One of the easiest things to do with your coffee grinds after brewing is to dump them into the garbage or wash them down your sink. With first hand experience, washing them down the sink WILL eventually clog your drain. Oils get stuck in the pea-trap and eventually you'll be washing those old grinds and a bunch of sludge off your kitchen floor!
As for the garbage, of course these will breakdown and compost on their own, but why throw out something that might be just as valuable as the coffee before you brewed it? Coffee grinds make up a great deal of our family's compost program, but even if you don't have a formal composting program in place, these grinds are packed with all sorts of benefits that can help you in any garden.
If gardening is not your thing, then there are some really interesting and overlooked uses for those grinds as written about at This Old House.
Thanks for reading.