My wife is a nervous traveler and takes medication to ease her anxiety, when we fly.  Even so, we usually opt for minimal connections.  My wife and I live in Florida too, so we are generally faced with a minimum of one layover, each way.

You don't mention a budget, or an originating airport, so it's hard to advise on flights.

A couple of discount airlines offer flights into Cork from the Boston and New York City areas (WOW and Norwegian Airlines) and Aer Lingus also offer some (seasonal) flights, non-stop from Orlando to Dublin.  Delta has wide-body jets that fly non-stop from Atlanta -- but, only into Dublin.  From Florida, most flights into Shannon go via Boston or NYC -- and United bases from Newark.  American services Ireland from Miami, Charlotte and Philadelphia. 

Glenbeigh is a VERY small place -- I actually have a friend that leaves near there!    -- but there are a LOT of terrific sites nearby, if you have access to a car.  It's part way along the Ring of Kerry and a short drive from Killarney and the Dingle Peninsula, so it's surrounded by glorious scenery and sights of interest -- castles, Round Towers and ancient Monastic sites. 

I really can't recommend NOT renting a car.  I live in Venice, Florida and I find driving here, during the 'Season', MUCH more stressful than driving in Ireland -- excluding Dublin city center (but, I've done that successfully, a few times!  ).  If you fly into Dublin, you could bus or train into Killarney and then rent a car.  If you fly into Shannon, you could rent there and spend the night in either Ennis, or Bunratty and then make the relatively easy drive to Glenbeigh and enjoy a brief stop in Adare (very picturesque) along the way.  The roads along that route are quite decent and the route is fairly straight-forward.  It's a bit much for a 'fresh off an over-night flight', though.

 Flying 'Open Jaw' (also called 'Multi-City') is very common.  You may find In/Out Dublin to be the cheapest option, though -- as it is the most common destination and offers more flights. 

Unless you have champagne tastes, that budget should suit fine for a Kerry-centric trip.  What interests you (other than family heritage)?  Scenery?  Architecture?  Ancient History?  Shopping?   . . .

If you flew into Dublin and spent a day seeing the sites, the train from Heuston Station takes you directly into Killarney, or continues on -- to the Kerry Airport.   A lot of the SNN flights from JFK, BOS or EWK are on 757 planes, that have a single, center isle, with three seats to each side.  Many Dublin flights are on A330 or 767 / 777 planes -- which feature 2-4-2 seating divided by two aisles.  Another important consideration is Travel Time (including length of lay-overs).  That can make flying to / from certain airports Less appealing over others.

For example, using your dates from Orlando, I found this $716 American Airlines deal: Departs MCO at 5:14 PM and stops in PHL at 7:45 and then, leaves PHL at 9:20 ( a 1 hour 35 minute lay over) and arrives SNN at 8:50 AM (Ireland time -- to you, it will seem like 4 AM -- only, in daylight!).  The return flight departs DUB at 9:55 AM (You'll need to be AT the airport about 7 AM) and arrives in CLT at 1:20 PM (Eastern Time).  After a 3 hour 5 minute layover, you depart CLT at 4:25 PM and land in Orlando at 6:06 PM.  That's a travel time of 10 hours 36 minutes GOING and 13 hours 11 minutes RETURNING -- which is pretty good.  Down side, for ME, is that planes used to SNN is a 757 (3 and 3 seating), but, the DUB to CLT flight is on an A330 (2-4-2) -- so you get a 'New, Best Friend' seatmate to SNN,     but NOT from DUB.  

A similar search to / from Tampa showed travel times in the 16 - 20 hour range . . .  BUT, there are LOTS of permutations and options that you could explore that could impact your choices, so the above is NOT a recommendation, per se -- merely an illustration.   

We retired to Venice a year ago -- moving South, from Gainesville, FL!    Needed to be close to the grand-children, don't you know . . .    

My wife's grandfather hails from a town about the same distance East of Killarney as Glenbeigh is, to the West, so I've come to know the general area rather well. 

I would agree with Bob about driving in Ireland, it would be quicker and less stressful than relying on bus transportation from Shannon to Glenbeigh. Also, you will be happy to have a car while staying in Glenbeigh. Looking at your travel dates Aug. 20 to Sept. 1^st, that will get you into Shannon on the 21^st, with about 7 days in Glenbeigh to include the races on the 26^st. I also agree with your cousin that you wont find enough to do in Glenbeigh, however you are so close to Killarney (about a 20 minutes drive), when she is at work you could drive into Killarney and do day bus tours from there. Also, on the road between Glenbeigh and Killorglin be sure to visit the Kerry Bog Village.

Flying out of Tampa I would recommend booking with Aer Lingus with connections out of Boston/Logan airport. Aer Lingus is a partner airline with Jet Blue so all reservation can be made with Aer Lingus for both airlines. Jet Blue to Boston and Aer Lingus to Shannon. Also at Boston/Logan airport Jet Blue and Aer Lingus use the same terminal. Again less stress for you and mom.

Another option once you leave Kerry you could return the car at the Kerry airport and take a bus or train to Dublin since you DO NOT want to drive in Dublin. In Dublin you will be able to once again do day bus trip tours around the Dublin area.