You can do most of the good day trips out of Dublin on the DART, Iarnrod Eireann or a single bus. Here are the ones worth your time when you do not have a car, in rough order of how easy they are.
Howth, fishing village at the end of the DART
Howth is thirty minutes from Connolly Station on the DART. Once there you have a working harbour with seafood restaurants, a cliff walk that takes about two and a half hours for the loop, and Howth Castle and its rhododendron gardens. Ivans for fish and chips, Octopussy's for a sit-down lunch, the Bloody Stream for a pint with a view of the harbour. Bring layers, the cliff walk is exposed.
Dalkey and Killiney, Ireland's Beverly Hills
South side of the DART, also about thirty minutes from the city. Dalkey village is fifteenth century, with a small castle and good food at Select Stores and DeVille's. Walk up Killiney Hill for the view that locals say is the closest thing Ireland has to the Bay of Naples. Bono and Enya live nearby. They will not be home.
Bray and Greystones, the cliff walk
Bray is forty minutes by DART. From Bray you walk the cliff path along Bray Head to Greystones, six kilometres each way along a path cut into the cliff above the Irish Sea. Get the DART to Bray, walk to Greystones, eat lunch at the Happy Pear or the Beach House, take the DART back. Five hours total including stops and lunch.
What is the best one if I only have time for one?
Howth in summer for the harbour and the food, Bray-to-Greystones any time of year if you want a proper walk. Dalkey if you have less than half a day. All three are direct from the city centre on the same single train line.
Glendalough, into the mountains
Glendalough is harder to get to without a car. The St Kevin's Bus from Dublin (St Stephen's Green to Glendalough) runs twice a day, takes about an hour and forty minutes, and gives you four to six hours on site depending on which return you take. Buy the ticket online at stkevinsbus.com a day or two ahead in summer.
Newgrange and the Boyne Valley
Bus Eireann runs the 100X from Dublin to Drogheda, then a connecting shuttle to Bru na Boinne (the Newgrange visitor centre). You need to book the Newgrange tour itself in advance, especially in summer; tickets sell out by noon on busy days. The whole day is about eight hours including travel.
How early should I leave Dublin?
Eight thirty for Howth or Dalkey gives you a full day with morning light. Nine for Bray-to-Greystones means you walk in the cooler part of the morning. Eight for Glendalough means you actually get a meaningful amount of time on site. Newgrange tickets dictate the schedule for that one.
Is the DART worth a Leap card or single tickets?
If you are doing one trip, single tickets are fine. If you plan three or more rides during your stay, get a visitor Leap Card at Dublin Airport or any newsagent. It caps daily spend and works on DART, Luas, and city buses.
What to skip
The Powerscourt Estate is hard to reach without a car. The hop-on bus tours go but cost forty euro and you spend more time on the bus than at the gardens. Wait until you have a rental car for that one.