how to get from Dublin to Galway?

Are you asking for a driving route? What is your definition of "best"? Fastest? Most scenic? Easiest for a beginner left-side driver?
The M6 appears to be a direct route, passing Athlone, which is on my list of places to visit. Castle, old pub and nearby Clonmacnoise.
The train goes the same way and also stops at Athlone.

ok, thanks for focusing my question! Has anyone taken the train from Dublin to Galway? Or is there a public bus? would a tour bus be better to actually see the sights?

There is a direct bus from the airport. There is probably also a bus from the city center, but I don't know why anyone would choose that over the train unless it stopped somewhere the train doesn't. The trains in Ireland, though somewhat limited in their range) are very good: clean, comfortable, inexpensive if you buy tickets in advance.
I am not aware of a tour bus that would take you one way from Dublin to Galway. You could hire a private driver, I suppose. There are tour companies that do single- and multi-day tours out of Dublin that include Galway. Rabbies and Paddywagon are the two main tour companies, but there are others. There are also tours departing from Galway.

I've taken the train from Dublin to Galway. It only takes a couple hours. It departs Dublin from Heuston Station, which can be reached via tram. The Galway station is an easy walk into the city centre.

Galway Ceannt railway station has this year been expanded from 2 to 5 platforms to cope with future expansion and extension of the Western Rail Corridor, and a new 2nd station entrance added.

Yes there are also buses from central Dublin-some of which are airport services.

Booked on the day at the station the fare from Dublin to Galway is €30.45, booked on line it is €16.59 amendable up to 60 minutes before departure.