Galway and the Cliffs of Moher are definitely too much for a day trip, but if you could overnight in Galway (stay at a B&B, don't stick to chains!) that'd be the way to go.

If you do have your own car, you can drive to the park at Killiney Hill and walk to the top from there, and then continue driving south to the Wicklow Mountains. Powerscourt, IIRC, is where there are absolutely massive gardens and a nice waterfall, I think there's a golf course somewhere around there too. Glendalough is also a great excursion, a little Googling will tell you all you need to know about it.

All the really great places to go are out of the city... it's a nice charming city, but it's not exactly full of museums and monuments.