Galway is safe it's a fairly small city full of collage students so most crime is drink related tomfoolery.
There is a train from Dublin to Galway but it's very expensive, €40 return I think. You can get a citylink bus for between $12 - $18. Bus takes 3 and a half hours max and their nice brand new buses.
What kind of stuff are you looking to see in Galway? Beaches, countryside, pubs, we've loads and loads of pubs, shopping, buildings.
You'll want to see the cladagh village, the Spainish arch, all the pubs, Silver strand.