We are planning to
1. Leave NYC friday night July 13 - Get in at Shannon Saturday morning.
2. What is there to see around Shannon Saturday and Sunday, July 14 and 15?
3. On Monday the wedding festivities start. One of the tours we will do together is The Cliffs of Moher, The Burren.
4. We will stay in the Dromoland Castle until Thursday morning July 19.
5. Then we are planning to drive to Dublin. What should we see between Shannon and Dublin (If we want to arrive in Dublin on Thursday Night)?
6. We will be in Dublin Friday and Saturday, what should we do there (Day and Night)? Attractions, Restaurants, Nightlife, etc.
7. We are leaving Dublin Sunday, July 22, morning or afternoon back to NYC.

What to do around Shannon depends on whether you will have a rental car. If so, there are:
Loop Head
Craggaunowen
Lough Gur
Bunratty Folk Park and Castle
Knappogue Castle
A bit farther afield:
Thoor Ballylee
Coole Demesne

Between Shannon and Dublin I would swing a bit north and see Clonmacnoise; perhaps Trim Castle if you have time.

Can't help with Dublin but plenty of others on this forum can.

And in the "farther afield" category there's Dunguaire Castle.

All of jaja's suggestions are great. Another stop on the way to Dublin could be Kildare - it has the National Stud (horse breeding), Japanese Gardens, and a well dedicated to St. Brigid all in one spot.

Dublin itself has loads of things. The Book of Kells in Trinity College, Guinness Storehouse, Kilmainham Gaol, etc. A good idea might be the Hop-on,Hop-off bus tour (get one with the live commentary, some of them are fantastic and sing!) to give you a great overview of the sites. Then you can pick and choose what to explore more.

Everyone knows Clare for the coast and Burren but there is East Clare as well. See the Shannon region web for ideas to fill your time in the area.

http://www.shannonregiontourism.ie/
http://www.shannon.ie/tourism.html
http://www.discoverloughderg.ie/

From the Clare Library site

7th - 15th July 2012
Willie Clancy Summer School, Miltown Malbay
40th Annual and Ireland’s largest traditional music summer school, where students from every part of the world attend daily classes taught by experts in Irish music and dance. A full program of lectures, recitals, dances (céilithe) and exhibitions are run by the summer school. Music sessions throughout the day in the local pubs.
Contact: Harry Hughes 00353 65 7084281/7084148
Email: [email protected]
Web: www.oac.ie & www.visitmiltownmalbay.ie

13th -15th July 2012
Kilkee & Loophead Maritime Festival
Web: www.kilkee.ie or www.loophead.ie