For Dublin Theatre go here:http://dublin.local.ie/general/theatre/index.shtml
I wouldn't say the Abbey is the best. Wouldn't know what to recommend, as it is a matter of taste.
Irish equivalent of the British Museum is the National Museum. Here is an excellent link, the Irish Times Visitor Guide:
http://www.ireland.com/dublin/visitor/museums/index.htm
There is an Impressionist exhibit that is supposed to be quite good running right now at the National Gallery of Ireland, website here:
http://www.nationalgallery.ie/
There are also excellent bus tours of Dublin available at the Tourist Office on O'Connell St.--a must for folks with limited time. It allows you to jump on & off the bus on YOUR schedule, not theirs!
Go here for your live view of O'Connell St!:
http://www.ireland.com/dublin/visitor/live_view/
And of course, you must have a nice relaxing rest after your shopping and sightseeing, at Bewley's on Grafton St--it is a Dublin institution!
The music calendar in Ireland runs from April-October, so you'll be a bit early for most music and dance events, including one of the highlights of the Irish traditional singing calendar, the Inishowen International Folksong and Ballad seminar, website here:
http://pages.prodigy.net/folkmusic/inishowenseminar.htm
Though you will be there for the great Corofin Traditional Music Festival, website here:
http://www.corofin.eu.com/
Both of the above has cream of the crop musicians.
An Goilin/Goleen Club website (out of date) here:
http://www.setdance.com/goilin/goilin.htm
Excellent choice, but you may miss it if you aren't in Dublin on the night.
When you land on the ground, get a copy of "In Dublin" magazine, as it has the most complete listings of trad music, with phone numbers for the pubs so you can call in advance to see if what is in the "What's On" section is actually on before you pay your taxi fare. Pubs in Dublin (besides the good suggestion of the Goleen Club and Hughes, which is at the back of the Four Courts) are O'Donohues, Slattery's, An Beal Bocht, and the Harcourt Hotel. All are found in or very near An Lar/City Centre, and easy to get to by taxi or on foot.
Since we don't know where you will be outside Dublin (you didn't say) it is nigh on impossible to direct you anywhere. I suggest you plan carefully in advance on where you will be each night, as 10 days isn't much time, and it is incredibly easy to waste it believing that going solo and being spontaneous will be at all productive and a good use of your travel budget. I suggest emailing the organizations/venues in advance of going if at all possible for a list of events in the areas you plan to visit. Since you didn't say if you would be going to the North (you should you know!) I haven't included much there.
CCE's website in Ireland has email contacts, very useful for contacting in advance for music events in local areas you might be visiting:
http://www.comhaltas.com
For fiddle music, there are tons of choices, but here are some I would suggest investigating.
For Donegal fiddle sessions, go here:
http://www.geocities.com/Athens/6464/session.html
For Sligo, go to the Coleman Irish Music Centre website:
http://www.colemanirishmusic.com/html/eventguide.html
For a list of Galway area music pubs, go here:
http://www.galway.net/galwayguide/pub/music/trad/
Belfast Gig Guide:
http://gigs.belfastcitylive.com/Belfast_gigguide/belfast_gig_guide.htm
Dublin Gig Guide:
http://gigs.gasta.ie/
Irish Times Dublin Live & Visitor Guide Page:
http://www.ireland.com/dublin/
An excellent on-line guide to Dublin is here:
http://www.houseofireland.com/dubguide/dubguide.htm
Even if you are leaving on the day, the St Patrick's Day Festival runs the 15th-18th, and the fesitval events are here:
http://www.stpatricksday.ie/cms/events.html
Temple Bar (Dubline) Gig Guide:
http://php.tbmc.ie/gigguide/
Heritage Sites website of the Dept of Arts, Heritage, Gaeltacht & Islands follows below--it is excellent! Be sure to get the Heritage Card if you plan on going to a number of the sites--it will pay for itself in no time. Their excellent website also has links for each site. I would definitely recommend Newgrange, The Garden of Remembrance (the wonderful Irish Writer's Museum faces the garden), Kilmainhaim Gaol, Trinity College Library and the Book of Kells, and Waterways Visitor Centre for in and around Dublin, easy to make it to all of them in two days time if you plan in advance. Newgrange is a half day minimum, and buses leave from O'Connell St. Tourist Office. Also, there are beautiful national parks to visit, if you get beyond the pale:
http://www.heritageireland.ie/en/frontpage.html