Flying Into/Out of Shannon: Cliffs of Moher, Experience Ireland's Beauty, Ideas?

Will you be renting a car or relying on public transportation?

Planning on renting a car.

Nearby - Skelleg Michael. Day trip - Giant’s Causeway. Otherwise, many great villages around SW IRE to explore and take in. Enjoy.

If you could extend it a bit, you could fly into Shannon and spend 3 nights in Killarney (loads to do around there) and then head back (by way of Dingle) to the Shannon area. You could stay at Dromoland Castle, visit the Cliffs from there and enjoy the "castle" experience.

Doolin is a good choice. Close to the Cliffs of Moher and ferry to Aran Islands. The town is known for traditional music in the pubs. It's close to Shannon. Easy-peasy.

IMHO, the Cliffs are vastly over-rated. There are better cliffs, better views, and innumerable alternatives. Sorry.

Try and extend the stay if possible as there is so much to do and see in the area. Flying into Shannon makes sense as you bare wanting to be in the west of Ireland.

Bunratty Castle and Folk Park is virtually next door to Shannon Airport. The Folk Park is grreat fun and tas well as going round the castle, you can attend a medieval buffet there too. There's plenty of choice of accommodation near the castle too.

The Cliffs of Moher are best seen from the sea - do this from Doolin and you can combine this with a boat trip to the Aran islands. You may also sea puffins from the boat as they nest on top of a sea stack near the cliffs.

My two Euro cents:
Most here advise against driving immediately upon arrival unless you're already experienced with driving on the left and are certain that you'll be well-rested after your long overnight flight. The suggestion to head over to Bunratty after collecting your rental car is a good one - it's only 10 minutes from the airport and spending your first night there or in the vicinity would allow you to get a good night's sleep before setting off on the local roads to begin your driving adventure.
Basing in Doolin would be my suggestion too, and if it's a sunny day I'd suggest driving up the coastal road (the N67 on the map) from Kilkee. It's one of Ireland's great scenic drives if the weather cooperates - lots of pretty seascapes and photo ops along the way. There's a very pretty cliff walk at Kilkee that would be a good leg stretcher, and the town itself is a cute place for a pit stop or a cup of coffee along the way.
Seeing Moher from the visitor center is a recipe for disappointment IMHO. The crowds alone can be off-putting, and the view frankly isn't as dramatic from the top as it is from the water ... ie on a tour boat or ferry from Doolin.
From Doolin the Burren is only a few minutes away too - and is worth the detour.
Adding a few days is always a good idea. In your case it would allow you to explore some of the beautiful Connemara area north of Galway before heading back to Shannon for your flight to Paris.
Booking a multi-destination (open jaw) flight plan would streamline your itinerary and preclude having to backtrack in order to catch your return flight. Might be more cost-effective too. Worth checking at any rate.

Thank you for the responses already. Now plan on staying a full week ;) Bunratty Castle looks perfect for a day and that day trip to see Gian's Causeway looks interesting. Oh my. So many cool things. Also, looking like there's a feel there are better Cliffs (by better, assuming a little less traveled ;) -- what would those be? Also, am fascinated by idea of seeing castle ruins too. And, other than by boat near CoM, are there other places to commune with the puffins? Any place people really love for a long sit, like people watching, absorbing the vibes, looking out on a peaceful spot, sketching, drinking a coffee, having some soup -- that sort of thing?

Also, any bookstores to add?

We've always enjoyed the Cliff Walk at Kilkee and frankly prefer it over Moher due to the fact that we always seem to have it all to ourselves. There's a very pretty stretch of sandy beach at Kilkee too, and we usually make it a point to do a pilgrimage to the sites associated with the actor Richard Harris ... who lived along the road leading to the cliffs parking lot. There's a little memorial to him just behind the Diamond Rocks cafe which is where the cliff walk begins. He was apparently a local athlete of some renown in his youth.
The cliffs extend along the entire N67 route and there are views all the way to Moher (and beyond) - just need to pull over and walk across the street.
Driving in Ireland (once you get off the Motorways) is always a lot slower that the mapping apps would have you believe. I'd add another 30% to your estimated drive times to account for slow moving local vehicles, sheep, cattle, etc. along the way.
For info, the Giant's Causeway is a 5 hour drive from Shannon airport. Not exactly a day trip.
In the short amount of time that you have I'd stick to the area around Doolin and Connemara - there's plenty there to keep you occupied for a week. The Sky Road scenic drive just west of Clifden is one of Ireland's best. The drive from Galway up to Clifden will take in Connemara National Park, Kylemore Abbey, and the atmospheric little village of Cong which is where "The Quiet Man" was filmed back in the 50's. Near Cong is Ashford Castle - a good place for afternoon tea and a wander around the grounds, and there's a "Hawk Walk" there that many on the forum rave about.
For an off the beaten path excursion I'd recommend the drive up to the Doolough Valley and it's famine memorial. Can do a google search to read about what happened there and decide whether it would be of interest.

Another pleasant day trip is to catch the ferry from the village of Cleggan (about 15 minutes from Clifden) over to the island of Inisbofin - one of Ireland's better kept secrets. It's a pretty little place and features an easy coastal walk that will take you by pretty beaches, old potato beds, a peat bog where you'll likely find some of the locals harvesting the stuff, and the ruins of an ancient church not far from the ferry landing. Might add that there a few very nice little pubs where you can recharge with a pint of Guinness before hopping back on the ferry for the return trip. The crossing only takes about 20 minutes as I recall.

day trip to see Giant's Causeway looks interesting

Robert has beaten me to this. Depending on where you are starting from, a day trip to Giant's Causeway isn't really practical as you are looking at at least 6 hours drive to get there PLUS time for stops... It's about 260 miles one way and driving in Ireland is slow. If Giant's Causeway is a real must see then you need to take two dayts and spend the night near there.

Robert has also suggested lots of other good ideas for Doolin and Connmara. You can make much better use of your time.

The Kerry Cliffs are also similar to Moher minus the crowds. You have to pay a farmer 8 Euros admission. It would also give you an opportunity to do Skellig Michael, which is one of our all-time favorite places, mainly because of the puffins being so bold with visitors. The beehives are also interesting, and maybe you're a Star Wars fan. May is ideal for puffins.

If the Cliffs of Moher are on your bucket list, you should absolutely visit them. They are iconic for a reason. Are there other (maybe more) spectacular cliffs in Ireland? Of course. Ireland has so much to see and do that you'll never see it all. Go where you want to go, see what you want to see. Don't lose sight of your plan for "a more leisurely soaking it up over fitting it all in."

Agree that the Giant’s Causeway is not worth the long trip up there unless you otherwise have time to see more of Northern Ireland. It’s fine but I was a bit underwhelmed. If it was in the United States I think it would be called Giant’s Causeway State Park.

I've done the Cliffs from the top as well as from the ocean. Both are well worth the trip. And both are totally different from each other.