Traveling without a car in early July

If you have 11 days in Ireland, you'll be able to see a lot, but not everything. The first few pages of the RS guide provide a suggested itinerary with destinations to add/subtract depending on how many days you have.

Ireland has trains, but mostly just between the major cities, so you'll be using buses to get to the smaller towns. Make sure to pack light, otherwise wrangling your luggage on & off buses can become not just inconvenient but a hazard to hurting your back, shoulder, wrist, etc.

Only you can say what really interests you -- for example, whether you're keen to see Northern Ireland or save it for another trip. You could do a triangle from Dublin to Cork and then to Galway, or the more northerly Dublin to Galway to Londonderry to Belfast. But probably not both.

Ireland without a car--you're comfortable with some bus travel, right?

I'd bookmarked a detailed trip report from Tripadvisor in which a couple traveled 3 weeks in Ireland sans car. TL;DR version: they used certain cities as hubs and took several day tours. It definitely has some good options for people who aren't game to drive on the other side of the road. Here it is: https://www.tripadvisor.com/ShowTopic-g186591-i88-k9151937-Trip_Report_3_Weeks_in_Ireland_without_a_Car_Sept_Oct_2015-Ireland.html

Epltd sketches out good potential itineraries. I would only add that if you don’t want to take group day tours (which really aren’t bad, you’re not following the pink umbrella with 75 of your nearest and dearest cruise mates), and are relying on public transit, you won’t be able to see as much in any given day.

I travelled for three weeks in Ireland without a car using a bus/train pass. I ended up mostly using the train to get to bigger cities then guided bus tours to see the sights. If you don't have a car the bus tours are really the best way to get to many of the sights. You can find companies that use smaller buses that might be more enjoyable if you don't want to travel with a group of 30 or more. The main places I stayed for several days were Dublin, Killarney, Galway and Belfast (four days each), and this worked well to access the guided tours. Kinsale and Kilkenny are also worth stopping at, but I would not use Cork. From Killarney you can take a small tour that will take you across the lake in a small boat and then get a jaunting car on the other side to go through the Gap of Dunloe (or you could rent bikes and ride through). From Galway you can take a bus to get to the ferry that goes to Inishmore then you can choose to rent bikes, take a jaunting car, or get a van tour from a local. When travelling by public transit 11 days is not as long as it might seem as you are at the mercy of their schedule, and buses in particular are at the mercy of traffic. So it is best to base yourself in a town for a few days so you aren't wasting a lot of your time on a bus or train travelling to the next town. So, what I did was I spent my first four days in Dublin then got on the train and went all the way to Killarney (over four hours) and after having done a lot of walking in Dublin the train ride was a nice break. Then I spent four days in Killarney doing tours and seeing the town. I got back on the train to Cork and from there took a bus to Kinsale for a couple of nights then made a stop in Killkenny for a couple nights before going on to Galway where I spent four nights. Since I had three weeks you will have to adjust how many stops you make and how many days you spend in each place, and you may not be able cover as much ground as you want. If Dublin and Galway are you main interests then you may want to eliminate the southern portion and split your time between Dublin, Kilkenny and Galway, or Dublin, Galway and Belfast and fly out of Belfast instead of returning to Dublin.