Day/overnight trips from Dublin

I'd have advised against having Dublin as a base for the entire stay too, to avoid a hell of a lot of to-ing and fro-ing. Oh well indeed.

Would it make more sense to rent a car for a few days of the 10 day trip, and do something like Cork to Ring of Kerry to Galway, staying in hotels / B&Bs along the way?

Yes it does. You could knock that off in 3 days if you so chose although I'd take 5 days, because everything you've listed is on the other side of the country to where you're supposed to be based.

I'd hire a car in Dublin, drive down to Cork (watch out for the tolls on the M50, read up in advance), consider stopping off at the Rock of Cashel for 2 or 3 hours which is directly en route. You can do Cork for a day, then the Ring of Kerry for a day, Dingle Peninsular for the next day (you didn't mention it, consider it), then amble over to the cliffs of Moher (if possible take a boat out to really get the scale)on morning day 4 then do Galway in the afternoon of day 4. Finally, do more Galway in the morning of day 5, then in the afternoon drive back to Dublin.

Agree with Fwoggie. In July with long days it should be possible to leave Galway on Day 5 around 8pm and still be driving in good light, thus giving a full day in Galway. Leaving Dublin early (mindful of rush hour traffic) it should take 2.5 hours to get to Cork, so allowing for stopover in Cashel you could reach Cork by lunchtime.
Or take train to Tralee and hire a car there for the Kerry section and then train back to Dublin after a day or two. Same with Galway area. See if the €9.99 (single) tickets are available on the days you could travel.

What about a night up to the giants causeway, and you can stop at new grange on the way back?
OR

Also I'd stop off at Cahir Castle rather than rock of cashel, theres more to it.
Don't forget there English Market in Cork city its lovely.
With this much time I would drive right down the beara peninsula taking in
sea kayaking with the seals in adrigole bay

Trip to Garinish Island to the Spanish Gardens http://www.garnishisland.com
Picninc at the ewe experience sculpture garden http://www.theewe.com
Journey on the only cable car in Ireland to Dursey Isalnd http://www.durseyisland.ie/cable-car-timetable.html

Eat fresh crab in castletownbere fishing port town and local milleens cheese
visit Eyeries one of the most beautiful villages in Ireland http://www.eyeries.ie
visit historical dereen gardens http://www.lonelyplanet.com/ireland/county-cork/eyeries-to-lauragh/sights/parks-gardens/derreen-gardens on your way to Lauraugh/Kenmare
You'd have all this done in 4 days cos its all relatively close together. this region is much quitter in terms of foreign tourists but has plenty of B+Bs/hotels and restaurants to cater for tourists.

If all else fails and you do end up going to the south of the country make sure to check out either Kinsale or Kenmare (if not both) and find a good sea food restaurant (unless your allergic ;)

Thanks for the replies. I don't know that I could leave the family behind for 5 days. I'm going to try for 3 days. So, given that, what would you see in that time?

I'm thinking possibly Cork, Ring of Kerry and... Dingle Peninsula? Galway?

Cliffs of Moher we may see as a family day trip from Dublin on one of the rail/bus tours. The one tour I found visits Cliffs of Moher, The Burren, and Galway Bay.