Where should we go from Dublin?

I would recommend taking the bus to Galway. There is a Dublin/Galway bus service, it takes about 3 hours to get there. Galway is a lovely, small city with fabulous restaurants - Aniar has a michelin star, Kai is delicious all the time, Rouge is very good french, The Huntsman has a great wine list and a wonderful sampler plate; McCambridges is fun for lunch… many of the other restaurants listed on trip advisor are not as good, but become popular because they are recommended to tourists. Also there should be traditional music in many places in Galway - Monroe's features great music - which you may need to buy tickets for; Cranes has locals nightly, Tigh Neachtin's is a cozy pub with music and so is Tigh Coli. (I lived in Galway from November - May - probably way more music in summer months in other places as well) From Galway you can arrange for a trip to the Aran Islands for a nice day of walking, or even see about taking a bus to Doolin and connecting with the walk along the top of the Cliffs of Moher. For that matter, you could probably catch the bus to Doolin and stay there for a few nights, but it's very small town.
Hiking in the Wicklow Mountains would be lovely, but it is tricky to get to the trails without a car. You can bus or train in to towns nearby, but if you look it up you are often kilometers away from the national park there.
Brugh na Boinne (aka newgrange) would be a great day trip from Ireland, interesting megalithic site - plus Hill of Tara where Daniel O'Connell gave a famous speech.

Well, hope this is helpful!

I would consider a Bus to New Grange and maybe stop off in Trim to see the castle, also a bus to Kilarney national park and stay a evening or two and see the Muckross house and lake, lots to see and do in the area, or maybe bus to Shannon and visit the Folk Park and the banquet in the Bunratty Castle in the evening (I really enjoyed this). In my own humble opinion, I found the Cliffs of Mohr interesting but only for about an hour and I found the Rock of Cashel to be rather underwhelming.
If you want a truly amazing experience travel to Ashford Castle and spend all three days there before heading back to the airport, this is an experience that will last a lifetime, be sure to take the class on falconry!

Galway is an easy train trip from Dublin, and from there you can do day trips to see the Cliffs of Moher/Burren area, Connemara region, or get over to Inishmore. Some people will disagree with me, but Galway was my favourite town. It's not too big, but has lots to see and do, and it's easy to find a guided tour if you don't have a car. Killarney would be a longer train ride, and it's more touristy, but like Galway it's easy to get a tour from. Kinsale is a nice little town, and is known as Ireland's gourmet capital, but to get there you have to take the train to Cork, then a bus. Kilkenny is also easy by train, and from here you could get down to see the Rock of Cashel by train/bus.

Thanks for the advice! Any suggestions on where to stay in Galway? We won't have car, prefer inn or small hotel...

We stayed at the Four Seasons, in Galway a B and B recommended by Rick Steves. We liked it very much (Cautions, the bedrooms are upstairs (no elevator) and the shower is small. The breakfasts are fantastic, the location is great, the hosts are friendly.

I also recommend Killarney. The scenery is beautiful, and the Gap of Dunloe tours are fantastic. You can hike in the National Park and see the Muckross House and waterfall right from town, there is a local bus. We stayed at the Old Weir Inn and liked it very much. We took a Ring of Kerry Tour from Corcoran Tours. It was a 16 seater. We wanted to stay 2 nights in Kells on the Ring of Kerry. Corcoran tours picked us up at our hotel, and dropped us off at the B and B in Kells. Then 2 days later picked us up across from the B and B took us on the rest of the tour, and then dropped us off at the train station for the train back to Dublin (3 hours).

Thanks for the suggestions, everyone! We went from Dublin to Galway by train and loved it! Stayed just 3 nights, at a convenient and comfortable B&B, Cappa Veagh, in Salthill. Hosts were so welcoming and helpful! It was a nice get-away and we did day trips without a car: Took a bus tour to Cliffs of Mohrer--spectacular. Took the ferry to the Aran Islands--what a trip back into time and so lovely. Spent a day walking the promenade into Galway, shopping at local stores, sightseeing, and enjoying a wonderful dinner at Oscar's (seafood). We will return to the West Coast of Ireland and hope to visit other parts of the country. The people were so friendly, we enjoyed excellent food (farm to table), and the landscape is gorgeous.