Galway is famous for quite a few historical sites, but the town itself isn't that big. Sometimes it's called a city, but it would be very small in comparison to any European counterparts.
Some of the most well known places are probably big let downs; Eyre Square is literally just a (small) park, The Spanish Arch is literally just a (small) archway.
However, the museum has free entrance and some interesting exhibits, and Salthill would be lovely if it's a nice day (when I went it was incredibly wet and windy - typical West coast weather!).
It's very popular for its nightlife (and food) and has some places (like hostels) you could stay while on your way around the surrounding countryside.
Cathedral, near the University
Eyre Square
Galway City Museum
The Spanish Arch
