Good locations for lodging for a golf trip, and a non-golfing week too

Since areas with a wealth of natual beauty and smaller towns are key criteria for both of you, then Connemara in County Galway would be my first choice. I consider it the lovliest area in all of Ireland. Clifden and the village of Leenane (the latter with a superb eatery for lunch and dinner, the Blackberry Restaurant: www.blackberryrestaurant.ie), Oughterard and the larger university town of Galway pretty well fill out places to stay there.

County Kerry and the Dingle Peninsula would be my second choice for scenery, mountains and a collection of small towns with abundant B&Bs.

Since I'm not a golfer I can't help you there, though there is a course in the village of Doolin in County Clare that abuts the ocean. Doolin is also the country's capital of traditional music. Three or four pubs there offer it nightly during the summer.

Hi Camy, here’s a link that might be helpful: https://www.top100golfcourses.com/golf-courses/britain-ireland/ireland. Also, consider flying into one airport and out the other (Dublin / Shannon).

There are several world class courses in Northern Ireland. A Google search will likely turn up a number of companies that provide package tours (some all-inclusive no doubt), which might be attractive to your husband if he wants to cover a lot of ground in a short amount of time, as well as having guaranteed tee times at some of the more famous links courses - some of which may need to be booked weeks or months in advance.
After his week in the north it would be a relatively simple matter for you to fly into Dublin, hop the train to Belfast (about 2 hours), and join him there for further explorations of the north, as well as a side trip over into Donegal and Connemara as has been suggested.
Alternatively, he could join you in Dublin from where you could both train over to Galway, collect a car there, and explore Connemara at your leisure.
With only 9-10 days of discretionary time at your disposal I'd resist the urge to try to cover too much ground, but rather concentrate on one particular corner of Ireland, and Connemara would be my choice too.

My wife and I did your trip about 10 years ago. I flew into Shannon with 7 other golf friends and we played 6 courses in SW Ireland over 7 days. We stayed the first few days in Killarney and played Tralee, Waterville and Old Head. Then went up the west coast to stay in Ballybunion and played Lahinch, Ballybunion and Doonbeg (now sadly a Trump property). Following our week of golf, some wives flew to Shannon to join us for a week of independent travel. I hope your husband has at least 3 other golf friends to join him. I used a US based golf travel service SGH Golf to book all tee times, hotels and transportation. We booked our own flights. It was a fun time, especially the links golf. We were lucky and played in rain only one day. September could be cooler than our June dates. Good luck? 🍀

Thank you all for your replies. The trip planning is progressing and at this point my husband is mapping out his itinerary for 5 days of golfing, and thinking of staying on the east coast closer to Dublin, but not in Dublin itself. He has an interest in several courses an hour or so north and an hour or less south. Can anyone suggest a good home base for that that is not necessarily in Dublin. He will have a car and is not interested in driving in the city. A great homey B & B or Inn would be perfect for him. Our plan after his 5 days of golfing is for me to join him and then we will be staying in Bandon (near Kinsale), Kenware, Dingle, Doolin and Galway over the next 11 days. He will probably golf one day near Dingle as we will be there 3 days. As someone suggested we will visit the Connemara at least 1 day. Thanks again for all of the tips and suggestions!

Hi Camy, you said that you were reading Rick Steves' guidebooks. In his Dublin chapter, does he say "if you rent a car at the airport, what town is more convenient to sleep in"? I know the guidebook I bought for my 2010 trip did.

HI Mary Pat! Thanks for your reply. I looked in the index and also the Dublin chapter and didn't see anything specifically regarding another close town to stay in. However, another helpful person on this forum answered another questions that I asked regarding Gallway vs Dublin, and he suggested Trim as a last night town to stay in. I looked it up and RS does in fact recommend this town for a 1st or last night stay. Only 30 miles or so from the airport and home of the famous Trim Castle. I am looking at accommodations right now...seems like a great little place to stay without all the hassle of having to take a bus into Dublin with golf gear and our luggage. Some pretty nice historic places to stay!

One of the prettiest courses in Ireland is the Old Head at Kinsale - about a 20 minute drive from the town.
Kinsale itself would be worth at least a couple of nights (or one full day) to both enjoy the golf course and to be able to explore the pretty little seaside town in the morning and later in the afternoon. It can get pretty clobbered with day trippers around midday, but things clear out dramatically once they've departed.
The seaside walk from the town down to Charles Fort is a particular favorite.
Just about everyone on the forum who's ever been to Kinsale seems to have stayed at The Old Presbytery (us too) - a very nice place. Would be a good bases for a couple of days.