While in Cork City?

genajie, it will be helpful for those trying to help to know whether or not you will have a car? Where will you be coming from, and where are you heading to, after Cork?
Give a little background please?

Very sorry I didnot make myself very clear last time.
No, we dont have a car, so we are going to rely on public transportation. We are going to fly to Cork and do a self-guided bicycle loop tour around Cork (Cork-Gougane Barra-Ahakista-Baltimore-Clonakilty-Cork), then fly out from Cork.
we have 3 days before the biking tour so was wondering if we should spend sometime around the city or cities/towns in the neighborhood. Any recommendations?
thank you!

That tour covers the southwest, so Mizen Head, the Bantry House and Lough Hyne must be included. Maybe you could take a one-day bus tour to other parts? County Kerry highlights would be very common, but see if you can find a less crowded route to the east of Cork that includes Ballymaloe and Middleton.

Do spend one full day in the city itself wandering the island centre and having lunch (buy some cycling provisions too) at the English Market.

Blarney, Cahir, and the Rock of Cashel are other possible day trips from Cork. You could probably easily find a bus tour that covered Cahir and Cashel.

Rick Steves and co also like Kinsale (1 hour by bus from Cork) and Cobh (half hour by train); both towns are covered in the book.