Via Motorways M1,M50,M4,M6, City to City, 230 miles,3hrs 40mins with tolls of 7.70 euros.
The inland route is 190mile but takes about 5.5hrs
Yours not going to have much time for sightseeing if you will be driving that much.maybe take a night off Dublin and it to Galway.
There's your proposed route - http://goo.gl/maps/lLHqs
As suggested by rojo81, you should allocate a second night to Galway, otherwise it is hardly worth your effort to go there for just one night.
When leaving Belfast, make sure you have Euros with you as you will have a number of tolls to pay - one on the M1 (€1.90) Belfast/Dublin, two on the Dublin/Galway motorway (€2.90 and €1.90) - the toll booths will only accept Euros, change is given. On your return journey from Killarney to Dublin, you will have one toll to pay on the M7 (€1.90).
As you will also be travelling on the tolled section between Jct.6 and Jct.7 of the M50 around Dublin on your way to Galway, there is a toll of €3.10 payable. However, there is no toll booth on the M50, and so you need to pay the toll at a Payzone outlet - http://www.payzone.ie/
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