Seeing as I'm from Dublin, I might as well chime in. Can't speak to Scotland as I haven't been as an adult.
Things to do in Dublin - pubs, lots of them, St Stephen's Green, Phoenix Park, Christchurch (especially the vaults), St. Patrick's Cathedral, Dublinia (a medieval/Viking museum), National Museum/National Art Gallery (both free), Natural History Museum (a Victorian era stuffed animal museum. Somewhat bizarre, but one of my favourite places.) I always got a kick out of knowing that I was walking the same streets as Wilde and Joyce. If you're a fan of Ulysses, various locations from the book are signposted. I'm sure there's probably a guided tour. Guinness Storehouse and have a free pint in the Sky Bar. Temple Bar is a cobblestoned, trendy, arty area. Very popular with tourists. There's also a touristy Viking amphibious vehicle which goes down the Liffey. There's a wax museum. There's a leprechaun museum (
), obviously I've never been. Go out to Howth, a fishing town just on the outskirts and get fish and chips from Beshoff's. While you're there, take the Howth Coastal Walk around the peninsula.
Great bit of advice for Dublin, but applies to most cities, is take one of those hop on/hop off tourist buses, so that you can see quite a bit, get your bearings and know where particular places are.
After Dublin, visit Newgrange. Then head north (Belfast probably has a ton to see too, never stopped there) to the Giant's Causeway and the Carrick-a-Rede rope bridge (On a clear day, you can see Scotland) Then head west - visit the Aran Islands, can be full of tourists, but it's my favourite place in Ireland. You can rent bikes there, or you can take a guided horse carriage (bikes allow you to explore more) to Dun Aenghus.
Back on the mainland, maybe visit the Cliffs of Moher, the Burren (I've never been to either) and then drive the Ring of Kerry (Killarney is the popular tourist stop) or the Dingle Peninsula. Maybe Blarney Castle (extremely touristy, most Irish people roll their eyes at the whole thing) There are plenty of other castles to visit. The Rock of Cashel in Tipperary, in particular, looks stunning. Other places that come to mind: Dunmore Caves in Kilkenny. Powerscourt House and Gardens in Wicklow. Glendalough is another common tourist stop. There are also plenty of other places to stop, small coastal towns and the like. I'm just pointing out some of the bigger attractions.