By plane
You can reach Athlone by landing in Dublin or Shannon. Athlone is just 90 minutes from either Dublin or Shannon Airport. Citylink bus company connects Dublin Airport with Athlone via it's Galway route. The bus stops at Arcadia about 15 minute walk north from the Civic Centre. The stop is a bus shelter.
By train
Athlone is on the Dublin to Galway/Westport line. The train station is located ten minutes north of Church St.
By car
The M6 motorway connects Athlone with both Dublin 90 minutes and Galway 60 minutes.
By bus
Athlone is serviced hourly by Bus Eireann from either Galway or Dublin during daylight hours. The bus station is beside the train station. I think there is a daily bus to Waterford, Sligo, Cork and Limerick. Within the town their is a local Bus Eireann service serving Monksland on the west side of the town.
Citylink multistop service from Galway to Dublin also stops in Athlone, at the B&Q store.