Hi Tammy,
How exciting that you are coming to Ireland!
Both Dublin and Galway are great places to base yourself for day trips - from either of these you can take great day trips that all have something different to offer.
From Dublin, I would definitely suggest going out to Howth and seeing the beautiful Howth castle and ruins of St Mary's Abbey.
A lovely day trip from Galway could be anywhere in Connemara national park, for example Clifden or Letterfrack. Connemara is easily accessible with
public transport from Galway.