Limerick is easy to reach from Shannon Airport (~20m drive by Taxi, 45 min by bus), Cork Airport (~2.5 hours by bus), or Dublin Airport (~2.5h drive / bus). Bus services provide convenient links to both airports. Details can be found below.
Ireland is served by many different airlines. Dublin Airport is the largest airport and thus best connected. However, there are good connections available from the USA to Shannon Airport on the West coast which is nearby Limerick and not too far from Galway. AerLingus is the national airline. Shannon, Cork and Dublin are all well connected with AerLingus and also with RyanAir.
A viable option is to plan a connection through one of the airports in London (Heathrow, Gatwick or Stansted), all three of which connect well with Irish airports. However, do plan for sufficient transfer time – the minimum for Heathrow is 1 hour, for instance (without any delays).
Ireland has several bus operators that serve the airports. Bus Eireann is the national public bus operator. Private bus operators include CityLink (to Cork/Galway), DublinCoach (an express service between Dublin and Limerick), EirEagle (a non-stop express service between Dublin and Limerick) and JJ Kavanagh (a non-express service between Dublin and Limerick). Details per airport are listed below. Please note that some services may not run at night. No reservations are necessary but online reservations may result in a discount for some services.
Irish railroad is oriented around the capital Dublin, and consequently taking trains between other cities may not be time-efficient. Also, when arriving at Dublin Airport, the easiest and fastest way to get to Limerick is with EirEagle bus service. Trains go from Heuston train station, and there are bus services from Dublin Airport to Heuston train station (~30-60 min depending on traffic).
Limerick has a good network of city buses that run regularly during daytime (and less regular during evening time). Alternatively, there are plenty of taxis around to take you to your destination. Limerick doesn’t have Uber drivers.