Car rental....we found the best rates and for ease of service ......we use Easirent. They are located just outside of the airport and will pick you up and drop you off. There are other good, reputable car rental agencies at the airport or just off site like Easirent that are fine. In the past we just found them more expensive and also very nit-picky on the details than Easirent. Regardless.....recommend an automatic and definitely get max CDW Insurance coverage INCLUDING tire and windshield coverage. So worth it for Irish roads.
As far as first day personally I'd skip Galway. It's a pretty city, but it's touristy, congested and its a city. Unless your really into city "stuff", which I would think would take a long part of the day to enjoy.....I'd decrease my overall driving times and just go to Doolin. There is plenty to do there. Excellent restaurants and pubs. Great trad music. If you're tired from jet lag and driving they have a super easy to use hop on hop off shuttle for a minor charge...Will take you all over including the Cliffs of Moher.. Which btw are best seen early or late in the day......less tourists and buses, birds more active etc. Also there are very affordable Cliffs of Moher cruises available. You could easily hop on an afternoon trip. They don't take long but are a lot of fun to see the Cliffs from under them.....impressive !
Heck, if you see the Cliffs on your first day after a more leisurely drive you either have more time to enjoy the next days drive to Dingle or you could from Doolin have a day trip to the Aran Islands.....take a horse cart or tour bus ride to amazing sights like Dun Anghosa. Just a thought. It really is beautiful.
If on Dingle you just want to do Slea Head drive that really is just one day. If you are also exploring other fun sights like the Oratory, a boat ride out to the Blaskets, and nice villages like Ballyferriter another day would be great.
If on a later day you are considering the Rock of Cashel....it and nearby ruins are very impressive and well worth a visit......but agree with previous poster it wouldn't be worth burning most of a day there and another day back......especially when in Ireland there is always amazing things to see and do near to where you are. If you do find yourself within an hour or two of Cashel it's well worth it.
No matter where you are there are great things to see and do nearby. So cutting back on driving is a good thing. The trade off of extra driving to see another "major" thing or sight isn't remotely worth it. In Ireland less is always WAY more....especially since wherever you are there are magical hidden gems close at hand....and ask locals for ideas.....they know amazing other things to see and do.....and allowing time to interact with the people is what turns an Ireland trip from a checklist completed into an amazingly deep life experience.
Have a great trip and happy anniversary !