Without knowing what your primary interests are (for example, coastal scenery, small towns, countryside, cities, historic sites), it's hard to make a list of recommendations. I suggest you buy Rick Steves' Ireland guidebook to help you make a list of "must sees."
The best way to see small town and rural Ireland is to rent a car. Note: the country may be small, but driving is slow. If you don't plan to drive, be aware that train service is somewhat limited in Ireland. You'll do better either combining train and bus or using the bus alone.
What's your price range for a B&B in Dublin?