There’s good generic advice about travelling around from Tourism Ireland here. But for particular points…
Bus and Coaches. There are direct bus links to Monaghan town from Dublin Airport, and also bus links from Belfast Airport, though this will require changes. Timetables may be accessed at www.buseireann.ie and www.translink.co.uk for the Republic and Northern Ireland respectively. The return journey to Monaghan from Dublin Airport costs €33.00.
Make sure and book your tickets but allow enough time to get through baggage reclaim and passport control.
There are also private coaches which are usually cheaper but not as easy to track.
The best thing about coaches in Ireland? Talking to people. But then that’s probably the best thing about Ireland!
Trains. There are no trains to Monaghan or in fact South and West Ulster but the rest of Ireland has a decent network to the main towns from Dublin, see here. And see Irish Rail for booking etc.
Cars. Again there is good advice here. The biggest difference in Ireland is that we drive on the left, in Monaghan and the countryside we have a lot of single lane country roads and have a lot of slow moving agricultural machinery. This may sound innocuous but from experience of driving on the Continent on the wrong side of the road (pun intended) this where you need to be careful if not used to it. Oh and we have a lot of roundabouts compared to the US and Canada, go clockwise!
Tolls. Driving on most of the motorways will involve tolls, typically €2 a go. There is one toll on the road from Dublin Airport to Monaghan so have your money ready. They do take cards.
The one exception to cash payments is the M50 around Dublin which has a Barrier-Free (CCTV) toll between junctions 6 and junction 7 to the west of the city. Even for rented cars you’ll need to pay online e.g. here e-Flow website M50 Toll by 8.00pm the next evening.
Car Rental/Car-hire. Again decent advice from Tourism Ireland here. The cheapest car-hire is usually to and from the airports. This site Moneysavingexpert.com provides more detailed advice and lists comparison sites.
And as Scott says below, check if your own insurance can or does cover you in Ireland. The rental companies here in Europe make money from quick up-selling to tourists who are just getting into holiday-mode.
One Monaghan tip: If you plan on travelling through Northern Ireland (to Belfast, the Titanic Museum, Game of Thrones Tour or Derry/Londonderry ) tell the car-hire firm, otherwise it invalidates the insurance (Bloody Brexit!)

If you have a tip or a question, please leave a comment and we’ll get back to you.

LOTS of Roundabouts! LOL!
If you’re planning Car-hire (Car Rental), ask about the insurance cost as it is way more than you expect in the U.S. Also make very sure if you plan to use your insurance or insurance through your credit card. It might not work as you expect.
Good point Scott. Check your own insurance and if possible get an upgrade as (in Europe) this is typically cheaper than the Car Rental companies. You don’t have to take their insurance if you are already covered
I think we were alright with the auto collision damage waiver we got from our credit card. Last time I think that saved us €126. (Scott, we talked about this once in the forum.) We obtained a letter from the credit card company stating we are covered for that, which seemed to work at the counter. We now also have an umbrella policy with our home & auto carrier, which they say will cover us for renting internationally. But, you’re right – you have to know it will work!