There's a car ferry from Portaferry to Strangford. It takes about 5 - 10 minutes, and they run every half hour (on the hour and half hour from Strangford, and at a quarter-past and a quarter-to from Portaferry. I got it to school for 7 years). If you're in Portaferry, have a drink in Fiddler's Green - you'll get a singalong in the evening, but not a hard-core trad session, but it's a very friendly pub and they do B&B too. That's my plug for the day!
I'm ashamed to say that I haven't visited the folk park in Omagh, so I can offer no opinion on it. There's a folk museum near Bangor, Co. Down, which is quite good.
Downpatrick is not the nicest of towns (personal opinion, I was at school there) but I do believe there's good music thereabouts. They have a folk club, and a webpage to go with it - click here. Try to get a drive through the Mourne Mountains as well - a beautiful part of the world.
There's a few sessions to be found in Drogheda (I will get up-to-date details from my Other Half who currently resides there). It's also worth visiting the archaeological sites therebouts, such as the Boyne passage tombs, Monasterboice etc. If you kick over a stone in that place you'll find something of archaeological importance.
If I can think of any more, I'll let you know.