Yes. Trains going from Dublin to Galway and from Dublin to Westport, go through Athlone. Both leave Heuston Station in Dublin. There are lots of services each day. Check the Irish Rail website from the link below.
There are various routes from Dublin to Athlone. The best is to go by the M4 motorway to Kinnegad and from there to Athlone by the M6 motorway. Going that way it is about 77 miles. That would be about 125 kilometres.
Depending on which Mayo station you are going to it will be between 3 and 3.5 hours approx. Some services require a change of train in Athlone but most are direct.
Dublin - South King - St Dublin 2, 40-43 Dublin - Henry St, 46-47 Dublin - Dundrum Town Center Dublin - Blanchardstown Centre Dublin - Pavillion Shopping Center Cork - Mahon Point Limerick - Crescent Shop.Center 34x Athlone - Athlone Town Centre Belfast - 3, Donegall Place Newbridge - White Water
That will depend on what time you are travelling, what day of the week you are travelling, and where in Dublin you are travelling to. To get to the city is about 2 hours, but once you reach the edge of the city traffic will get heavier, so the last part of your journey would be slower if you are driving. If you are travelling by train, that will not be a problem.
There is no train from Dublin to Waterville. You could get a train to Tralee or Killarney in county Kerry, the county that Waterville is in.
Killarney is approximately 4 hours by road or by train from Dublin
Connolly Station - just in the north inner city of Dublin
There is one in the Athlone Shopping Centre.
Athlone GAA was created in 1885.
The motto of Athlone is 'Urbes Stant Legibus'.
The Guinness Book of Records has Seans Bar in Athlone Ireland opening circa 900AD.The Brazen Head in Dublin has been around since 1198.
The Eurostar goes from Paris to London, it doesn't go to Dublin.