Just to the north of Cork are Cahir and Cashel. Cahir (pronounced "care") is the home of Cahir Castle, which is nice to visit. It was used in some scenes from the movie "Braveheart". Cashel has the Rock of Cashel, one of my favorite locations in Ireland. Both can be done in a day. Make sure you see the film they show at Rock of Cashel and visit the little museum.
Kinsale is to the southwest of Cork, on the coast, and Cobh (pronounced "cove") is to the east. Cobh was the last docking place of the Titanic before its voyage, and has a wonderful maritime museum. Midleton is beyond Cobh and is the home of the Jameson Whiskey distillery. Cobh and Midleton can easily be done in a day, Kinsale would take a separate day. Fota Wildlife Park is also near Cobh.
And if you are looking for something a bit different, on the road from Cork to Limerick you'll come to a town called Buttevant. If you turn west there and head out into the country you'll come across a donkey sanctuary. They take in mistreated and abandoned donkeys and care for them for the rest of their lives. It's a wonderful place to visit (and free). You'll also pass a huge 13th century castle on the way. If you'd like more information about this, please message me.