The National Botanic Gardens in Glasnevin, Dublin is a brilliant place to visit. It is free to enter the gardens and is a great place to go for a walk.
The gardens were founded in 1795 by the Dublin Society (later the Royal Dublin Society) and are today in State ownership through the Office of Public Works. They hold 20,000 living plants and many millions of dried plant specimens. There are several architecturally notable greenhouses (from wiki).
The gardens have a cafe and restaurant on site and plenty to see. Here are some photos from a recent trip.
By the way, it is right next to the Glasnevin Cemetery so you could always combine the two for a days visit!
There is a carpark on site but it can get busy at peak times. Location below.