While I think that traveling during the shoulder season is generally a good idea, be aware that you'll be there just when the weather (historically at least) can be quite changeable - with increased chances of rain and windy days. The days will start to get a bit shorter, too, so I think that getting an early start every day, with a lot of latitude built in to your itinerary is a good idea, ie your decision to stay 2 or 3 nights in each location is a good one. Be prepared for at least some rain, hope for the best, and know that rainy days won't be a total loss - they're why the Irish invented Guiness, convivial pubs, and warming peat fires after all.
The ferry out to the Aran Islands will be very weather dependent. Have a backup plan just in case the weather doesn't cooperate. That said, Connemara is spectacular - well worth several days of your time.
If your flights aren't yet booked you might investigate doing the trip as an open jaw, ie flying into Dublin and then home out of Shannon. Would save you a day of back tracking anyway.
Know that you could take the train from Dublin to Galway on day 3, then arrange for your car pickup there. Might be marginally cheaper to do a one-way rental from there back to Dublin rather than collecting the vehicle in Dublin itself. Try to avoid driving in Dublin center, ie pick up and/or drop off your vehicle at the airport or somewhere else on the outskirts of the city, especially if this will be your first experience driving on the left. There's lots of good advice on this forum regarding vehicle rental in Ireland - from reserving one with automatic transmission to detailed discussions about which insurance coverage to spring for. You'll need at least a medium sized vehicle to handle your party of 4 with luggage, and Ireland's sometimes narrow (and at times very narrow) country lanes can be a scary experience.
Other than Adare, another overnight possibility after your day at the Cliffs and the Burren would be Kilrush or Kilkee. There's a very pretty coastal walk just outside Kilkee which, if you're into that sort of thing, might be a pleasant way to end your day. It's not the Cliffs of Moher but it is quite nice...with lots of photo ops along the way. Could then catch the ferry across the Shannon the next morning from the little town of Kilimer for a very pleasant drive down to Dingle. Others have commented that the roads on the Kerry side are not in particularly great shape and therefore recommend driving around the (slightly) long way via Limerick. Can't comment on that except to say that when we did the route many years ago the road conditions weren't an issue.
Best wishes and good luck.
