I grew up in the South East and return frequently. Some of my favourite areas (in a semi-geographical order):
East Coast:
Wicklow Mountains - Old Military Road through the Sally Gap (R115) - stunning drive.
Glendalough - outstanding scenery and old monastic sites on a lake. Make sure to walk up to the old lead mines.
Wicklow Gaol for a bit of heritage.
Curracloe Beach and The Raven in County Wexford (photo included - where Normandy landing scenes in Saving Private Ryan were filmed). 30 minutes up the coast from Rosslare Harbour where both Stena Line and Irish Ferries operate to from Pembrokeshire. Great place for a long walk along the beach or through the forrest.
Wexford town - Old viking town which comes alive in late October for an Opera Festival and art exhibitions.
Lobster Pot Restaurant - near to Lady's Island. Fab seafood restaurant.
South Coast:
Kilmore Quay - Silver Fox Seafood Restaurant used to be very good. Hope it still is. Nice beach nearby.
Hook Head Lighthouse and peninsula.
Duncannon for another good beach. Perfect place to learn to kitesurf with Hooked Kitesurfing.
Ferry from Ballyhack to Passage East and then explore Dunmore East and Tramore (good for surfing).
Inland:
Borris - a visit to Joyce's Pub for a pint of the black stuff is a must. It's like stepping back in time. Time your visit for a tour of Borris House which is open on certain days from May to September. Extraordinary history of Arthur MacMurough Kavanagh.
Kilkenny is the pick of the large county towns with an outstanding castle. Nearby Thomastown and Graiguenamanagh are picturesque small towns on the River Nore and Barrow.
However, in my opinion, all of this pales in comparison to the rugged natural beauty and isolation of the West Coast. Places like the Dingle Peninsula in Co. Kerry, the Burren in Co. Clare, Clifton in Co. Galway, Achill and Belmullet in Co. Mayo. I still need to explore some of the remoter areas of West Cork and the entire Atlantic coastline.