Dublin - Beginning / End or All At Once

I suggest starting in Galway--even if you are flying into Dublin, there is a direct bus to Galway (better to fly into Shannon though). Spend your time in Dublin at the end. (My wife and I made a return visit to Ireland in May 2016, flying into Shannon and taking the bus to Galway.) With only ten days on the ground I might shave a day off of Dublin and spend it elsewhere. The best of Ireland is out in the countryside and smaller towns, though nothing against Dublin. For me the best of Dublin was the archeological museum and Kilmainham Gaol, Dublin Castle was also very good. The countryside is so much the best of Ireland, I would even suggest skipping Galway on this trip and spending time in County Clare, Dingle peninsula, Kenmare as a gateway to Ring of Kerry. Edit: And if you are renting, which it looks like you are, get the super CDW!

Are you renting a car? With only six days to cover all of those places it will be very difficult without a car. Galway was my favourite place, there is so much to see and do in the area I spent four days there. I would skip Cork and spend a couple of nights in another town from which you could do either the Ring of Kerry or Dingle Peninsula. If you are not renting a car then I highly recommend researching the train/bus schedules and then decide where to spend your other six days. Both Galway and Killarney are going to have the best train/bus access, and both have many companies doing guided day trips to what you want to see. If you haven't booked your flights yet I would also suggest flying in to Shannon and out of Dublin, or the other way around.

Good advice from VS and Anita. Would only add a few thoughts:
1. Coming from Chicago you'll probably be landing in Dublin (or Shannon) in the early morning after a long overnight flight. If you plan to drive, you should consider how well you sleep on airplanes and, in particular, whether you're going to feel up to the challenges of driving on the left immediately upon arrival. Might be better advised to bus or train over to Galway for a day or two to log a couple of night's sleep before getting behind the wheel, and thus begin your driving adventure away from the Dublin metro area. You won't be able to check into your hotel until mid-afternoon anyway, so why not put that time to good use by heading straight to the west of Ireland...less than 3 hours away by train or bus.
2. In July you'll be there at the height of the tourist season, popular destinations like Killarney and Dingle will be crowded, and accommodations will book quickly. You need to sort out your itinerary and make your bookings well in advance.
3. Connemara (where Galway is located) is one of the most scenic areas in the country. Day trips from Galway to Clifden and the Sky Road, the National Park, the Aran Islands, etc. can easily take up 3 or 4 days, and the area isn't typically as clobbered with (other) tourists as the more popular areas to the south.
4. The default rental in Ireland has manual transmission ( ie a left-hand shifter). Consider specifying automatic transmission if it's your first trip - it can be half-again more expensive than a manual, but will make life a lot easier for you in the long run.
Safe Travels.

Thanks for the replies. We are not planning to rent a car - we both dislike driving and aren't wild about navigating in a strange area on the left side of the road. Our plan is to take the train / bus to Galway and Cork and use them as bases for organized day trips to Ring of Kerry, Aran Islands, Dingle Peninsula and Cliffs of Moher before heading back to Dublin. Hoping this is a happy medium between doing everything on our own and spending the entire time with an organized tour group.