What radius mileage wise do you want to travel from Dublin since I can't ascertain do you mean the whole of Ireland or within a certain radius near Dublin. Foir the moment I'll take it you mean the latter but you can correct me if I'm wrong.
1. I would consider the top attaction to be Glendalough meaning the Glen of the Two Lakes. It is situated in County Wicklow perhaps an hour or less drive from Dublin. If you're driving it is worth heading over the mountains atleast one way to get there, just don't get lost and don't try it in the dead of winter if there is any danger of snow. Motorists have got stranded up there. You can return by a relatively flattish road which will connect with the motorway back to Dublin. You could head over the mountains by Glencree which has a German war cemetery & then over the Sally Gap. You could return a different route over the mountains by the Wicklow Gap. Bring walking shoes because the walking there is tremendous. By the south lake you've got a boardwalk & by the upper lake, which is even more spectacular, you have a well maintained path which brings you to a deserted mining village. There is a roundtower there, an interpretatuve centre, an old graveyard, a sprinkling of places to buy souvenirs & a nice hotel if you want to stay there or be fed & watered. There is also a very nice hostel there for those on a budget. If you travel the flattish road, stop at Roundwood (the road over the Sally Gap can also be accessed from here) & eat at the Roundwood Inn in the room next to the bar. It is run by a Swiss man and the menu gas a continental feel about it. The chicken & chips are to die for & the're not expensive. Alternatively across the road they do a very nice carvery. If you cannot drive, a private bus company called St Kevins Bus has a monopoly on the route but it won;t bring you over the spectacular mountains & you'll miss out on alot. If you want to overnight, taking this bus makes sense if you can't drive. Alternatively you could do a day coach tour which will bring you over the mountains. about 6 years ago
2. Hot on the heels of Glendalough is the pretty village of Enniskerry and nearby Powerscourt House. The village can be reached by public bus from Dublin, think it's the 44. It has nice eateries such as Poppies cafe, a few pubs & Summerhill House Hotel. Powerscourt Demesne is a short walk and the walk through the grounds up to the house is beautiful. The grounds have a golf club, the Ritz Carlton Hotel with a restaurant endorsed by Gordon Ramsey. The house itself has a tour of the house, beautiful gardens, quirky shops, a self service restaurant and an adjacent garden shop.
3. North of Dublin the landscape is less spectacular and flattish and the main attraction there is the archaelogical site of Newgrange. Also but far less spectacular is the Hill of Tara but don't do the latter without a guide because it will mean nothing to you.
4. The coastal Dart trip. Buy yourself a day pass & you can hop on at a number of stations in Dublin. Head south sitting on the left side. Get off in Killiney said to resemble the Bay of Naples & walk back by the coastal road to the pretty village of Dalkey for a nice meal & to browse the shops & maybe visit the castle. Hop back on the Dart heading southwards again. Hop off at Bray, visit the aquarium or have a nice meal. Walk the promenade towards the hill of Bray Head. Climb it or do the spectacular coastal walk to Greystones. Alternatively hop back on the Dart heading southwards and do the spectacular rail trip to Greystones which was a pretty village but whose coast has been partly ruined by an unfinshed marina. Hop back on the Dart & head northwards to Howth or Malahide. The former has a nice pier, stunning walks around Howth head and a plethora of restaurants. Buy some fish or maybe eat at the Bloody Forge beneath the railway station. Malahide is a pretty village wuth nice restaurants & a marina .Nearby Malahide Castle can be toured. It has nice gardens also & a model railway attraction. These are all des res areas. Alternatively when heading south get off at the ferry port & town of Dun Laoghaire. Walk the pier & coast, shop & dine in a nice restaurant. about 6 years ago
5. The Japanese Gardens, The Irish National Stud & St Fiachras Garden in Tully Co. Kildare. Also take a drive through the Curragh, famous for its horse racing and if you can divert into the military town of the Curragh Camp, it is an experiience.
6. Further north drive up to Carlingford Lough and on to Carlingford. If you've time to head over the border to the north & explore the Mourne Mountains all the better.
7. A tour of North County Dublin taking in Skerries & a visit to its windmills and a visit to a restaurant down its coastal road.
8. Mount Usher Gardens in Ashford, County Wicklow.
9. A walk up Three Rock Mountain if you can find it. On one side you will have spectacular views of the coast, on the other side the Dublin mountains. This is part of the Wicklow Way, a famous & lengthy walk through Dublin & Wicklow starting in the picturesque Marlay Park, much of it through mountains. It will eventually bring you on to a road which will lead you to the Famous Pub Johnny Foxes in Glencullen This can also be accessed by car from the Enniskerry road or by the rare public bus whose terminus beyond the pub is near the Wicklow Way. Decorated with old Irish memorabilia, it is famous for its restaurant, especially its fish. It also has night-time Irish emtertainment but since it's in the middle of nowhere it would be easier to see the latter in town say in Knightsbridge, the Arlington Hotel, near O'Connell Bridge.
10. Not technically beyond the Pale, as we call Dublin, is Phoenix Park. On Saturday mornings you can visit Aras an Uachtarain, the Presidents residence but you have to queue. Thinks it's on a first come first served basis. Famed for its deer, Phoenix Park is something like the largest enclosed park in an urban setting in Europe. It also houses Farnleigh House which houses visiting dignatories and be visited sometimes. Also often there are farmers markets there. The Park also houses Dublin Zoo, a great childrens playground, a visitor centre & the Wellington Monument.
11. Trim is a pretty village in County Meath with the spectacular ruins of a castle. You'd get to experience another county too.
about 6 years ago
If you've time to head south to the Medieval city of Kilkenny with its beautiful castle all the better. Further south on the south Wexford Coast is the very pretty village of Kilmore Quay with its thatched roofs, nice restauants & nearby beaches to walk. Visit the memorial to the Aer Corps airmen who died in a rescue when their helicopter ploughed into the high sand dunes at Tramore. Both these destinations are possible on a day trip from Dublin though you will be a long time driving. about 6 years ago
There's one I forgot. That is head out in the direction of Blessington. Visit Russborough House & do a tour, take on the maze.
Head back towards Blessington. Turn off for the Avon Ri Lakeside Resort. Prerequest a lakeview table & dine looking over stupendous views of the Blessington Lakes. Take a walk through the grounds down to the lakeshort & drink in the beauty of the scenery. You can book yourself in for self catering accomodation if you wish or just visit it as a day trip.
If you're feeling energetic prebook yourself in for some activities in the adventure centre there, both land based & water based. Only a short drive from Dublin you find yourself in scenery which would rival anything in Europe. You can also get a 65 bus there about once an hour. The walk to the resort is short.
If you want to do some shopping on the way back stop off in Tallaght and visit its shopping centre. You'll have a better choice of buses from there. about 6 years ago
There is one other place I forgot and that is Bohernabreena Reservoir, Glenasmole, South County Dublin.
The walking & views in this are are tremendous and you're barely out of the built up area of Dublin. Please view above link for public transport & car access. about 6 years ago