LEAP CARD and best apps for bus/train/taxi travel in Ireland?

It depends on where you're going, to an extent. If you're taking the train, I recommend booking ahead using the irishrails.ie website. Tickets come to email, but probably could also go to the TFI app? Advance fares save quite a bit, IME.
I found the TFI apps a bit of a challenge to use with the buses in the West.

Thanks. We will be using buses between Dublin and the Dundalk area, and then go by train to Galway, and use buses to smaller towns in the area.

You can also take the train to Dundalk from Connolly Station (train to Galway is from Heuston). I'd take a train over a bus most times. More comfortable.

I had looked into the train to Dundalk, but we will be flying into Dublin and going directly to Dundalk, so it appears a bus from the airport will be easier than going into Dublin to catch a train. On the way back to Dublin we could probably do the train. Does that sound right?

Yes a bus from the airport sounds sensible. It's an hourly service, 20 hours a day, Bus Eireann #100X.
Journey time 1 hour 20 minutes.
It makes 3 stops in Drogheda and 4 in Dundalk

Thank you!

Not sure if you have updated your question, but LEAP has covered the whole of Ireland from since it started. It is just that most visitors are only familiar with the LEAP visitor card for Dublin, partly because few people on the forum use the local bus services you can use it on. They either drive or use the private express bus services on which it isn't valid.

It isn't valid on Inter City rail services, just commuter rail services in the Dublin and Cork areas.

It is valid on all TfI supported services (like Dundalk to Newgrange) and on Bus Eireann local services, but not express services. So isn't valid on the Dublin Airport to Dundalk 100X service or the express from Shannon Airport to Ennis, but is valid on the Shannon Airport to Ennis local service. If in doubt check the BE fare finder- https://www.buseireann.ie/fare-finder
If a Leap fare isn't shown then it isn't valid.

In the Dundalk area it is valid on local services like the 160 to Newry or the 100 and 168 to Drogheda, also the 101 Drogheda to Airport.

Thanks so much, isn31C. These are helpful clarifications. We have leap cards from a year ago and can top them up via the app, but wanted to understand what would and would not be covered. We are determined to do this 3-week trip via public transport!
Thanks again.