10 day road trip of Western and Northern Ireland

I'm definitely in the minority, but I value driving time and variety more highly than the "base" concept with it's associated backtracking. We travel with just carry-ons and rarely spend three days in one place (Dublin, Belfast, Cork). Checkin/out is maybe five minutes. Driving from Galway to Connemara and back, then later driving to Sligo can cover the same country twice. For example, there's a day's worth of interesting things between Galway and Connemara, so I'd prefer to visit the places in or very close to Galway one day, then visit the far flung places on the way to a new lodging. In 2023 we did Belfast x3, Portrush, Dunfanaghy x2, Donegal x2, Westport, Clifden, Galway, and on.

Just my 2 cents...

Having recently done a condensed reverse version under a shorter time limit Dublin/belfast/GC/Donegal town/Dublin I think this is doable.

I understand the appeal of three night stays but I would suggest 2 nights in Belfast and make the most of the drive back. There are a lot of valuable places to see along the way. Last night close to the airport in Shannon to cut down on stress.

I visited sligo/galway/connemara on different trip so we didn't hit those recently. Sorry.

That said, if time permits I would suggest a few nights with a base in Donegal and a few nights on the coast (this board is full of a lot of suggestions to base yourself for the Antrim coast). I wish we had more time in Donegal - it was a favorite (slieve league!) and I wouldn't sleep on an afternoon in Derry as part of a travel day (or Grainan Aleach - great view).

How about flying into Shannon and out of Dublin? You'll pay a fee for dropping the car off at a different location, but you'll maximize your time. You should absolutely take as many days as you possibly can for this trip.
I agree with the other poster that the base and backtrack concept is not what I would do, especially for a solo traveler with a car who is trying to cover as much ground as you are and wants to enjoy the drive. Ask yourself why you want to stay in Galway. Is it to see Connemara? Then stay in Connemara. Same with Belfast. Actually, Belfast is one place where I think backtracking might not be a bad idea. Drop the car in Dublin and take the train or bus to Belfast so you don't have to deal with parking or with returning the car right before your return flight.
Take Uncle Rick's advice and make a list of the places you want to visit and estimate how long you need to spend at each. This should help you to plot your route. When you're on your own with a car, I think the single-night stopover is easier to achieve because you can treat the trunk like a suitcase and you're not waiting on anyone else. I'm certainly not recommending doing it every night, but I wouldn't rule it out, either.
Remember that with a car, you don't have to stay in a city. You can stay anywhere. For example, lots of folks on the forum recommend staying in Letterkenny to visit Derry because you can take public transit into town and not worry about parking (although there is parking at the mall). Once you have your route mapped out, zoom in on Google Maps (or the map feature on AirBnB) to see what accommodations are available along the way in a town you maybe never heard of.
I'm sure you'll have a great time. Enjoy!

If I’m reading your timing correctly, you’re planning on from 10 days (9 nights?) to 14 days (13 nights?), and 6 of those nights are already going to be in Galway/Dublin, so does that mean that you’ve got 3 to 7 nights for your additional bases? If you’re committed to 3 night stands everywhere, then it appears that you’re looking at either one or two places, besides Dublin and Galway.

We’ve used Shannon Airport on other trips but in the fall of 2018, we had a trip similar to the itinerary you’re considering. We did mostly 2-night stays at B&B’s, except for a couple of cities, and flew in and out of Dublin. Landing at the Dublin airport, we immediately caught a Citylink bus to Galway. We sat in seats at the front of the upper level, got a mini tour of Dublin as it passed through downtown on the way to Galway, and had a little post-flight nap enroute, as well. The double-decker bus had a restroom on board.

After Galway (without a car), we rented from Enterprise and drove clockwise to the top of Northern Ireland, then down to Dublin, where we turned in the car for our final nights. We had three weeks. After Galway and a stay in Inishmore (flying Aer Arann instead of using more weather-dependent ferries), we stayed in Clifden (great day trip options), then on Achill Island (with a bridge to/from the mainland), then a rural B&B a short distance northwest of Ballyshannon, up to Dunfanaghy, stopped off in Londonderry one afternoon during a festival, over to Bushmills (great base for Antrim Coast and area), then Belfast area, and finally Trim (for access to Boyne Valley prehistoric and historical sights), before Dublin for our last few nights, where we didn’t need or want a car. All this kept us north of Shannon, but we toured southern Ireland (and Ennis, flying in/out of Shannon) on our next trip, in 2023. With just 1 1/2 to 2 weeks, for the northern trip, I’d skip Inishmore, Achill Island, and Dunfanaghy.

I think it makes more sense to drop the car in Dublin (even better if you could drop it in Belfast but I understand why that's not possible).

Luckily, Northern Ireland is very small so most of it is within day trip distance of Belfast. Some of my favourite places for day trips: Enniskillen/Lough Erne in Count Fermanagh (this will be the longest day trip but worth it), Causeway Coast, Rathlin Island, Ards Peninsula (lots of nice places along Strangford Lough), and the Mourne Mountains. I don't see the Mourne Mountains mentioned much on here but I think they are one of the best parts of NI.

Lots of great historical sites and city life in Belfast- I really like the Ulster Museum, Titanic Museum, and St George's Market. I also recommend doing a black taxi tour.

Good luck with the planning!