I've got a gig at the Old Folks Home this afternoon, and I'm planning on
"Whisky In The Jar" (a perennial party song, about worn out in Celtic Trad circles but a big hit among those not burned out on it - especially after a couple of brews!)

Another rousing tune would be "Risin' of the Moon" or "Wearin' O' The Green" (they share the same tune, essentially)

"Finnegan's Wake" , a lighthearted ditty with a catchy tempo, rousing verse, and funny lyrics.

"Star of the County Down" - pretty, bouncy, romantic.

More sentimental; (good for the "senior" crowd); "Danny Boy", "Wild Irish Rose", "Mother Malone"...

If you're into vocals and can hit a full 3 octaves, (or at least 2 3/4, as I recal) try "Kathleen Mavourneen" just for fun!
I like to attempt it in the shower or driving (alone) as a vocal exsersize to push my range. Octave jumps, lots of "accidentals", slurs, and all that fun old stuff!

To close, I do "Parting Glass" and/or "Amazing Grace" on my low "G" open wooden flute.

Irish music is phenominally rich in history, tradition, and spirit - it's a lot of fun all around, really - and once you get into it you might not find it so easy to leave it alone!

Happy St.Pat's, all!