I've stayed at The Four Seasons B&B on College Road (mentioned in Rick's Ireland book) several times. It is a 10 minute walk from this B&B to Eyre Square and the bus/train depot. The hosts, Eddie & Helen Fitgerald, are very nice and they feed you way too much for breakfast.
Galway is relatively small, so it's pretty easy to get around on foot and a cab ride to just about anywhere else doesn't cost that much.
If you want to be right on the square, the Merryck (formerly the Great Southern) is the big, fancy hotel. Skeffington Arms is also on the square and there is a Jurys hotel over near the waterfront.
Check TripAdvisor.com if you want to read traveller reviews or see photographs of these hotels or B&Bs.