County Westmeath (Contae na hIarmhí) is in the East Coast and Midlands region of Ireland, and historically part of the kingdoms of Meath and of Leinster. It's about 90 min travel northwest of Dublin.
From the 1st to the 12th century AD, the Kingdom of Meath (Mide, meaning "middle") was a broad belt from the Shannon to the east coast of Ireland. Then the Normans arrived and began to subdue the Gaelic kingdoms and parcel the land into shires or counties. Meath was absorbed into the vassal kingdom of Leinster, to re-emerge as County Meath. In 1543 this was subdivided into an eastern part very much under Tudor English control and becoming the modern County Meath, and a western part that had remained more Gaelic. This "Westmeath" contained what is now County Longford, but that became separate in 1586, leaving County Westmeath on its present footprint. Local government reforms in 1898 completed the transition from wopentakes and baronial demesnes into the 20th century governance of county road building, libraries and schools.
Westmeath is lowland and agricultural, with easy transport to Dublin. As if it didn't have enough water courses, early in the 19th century the Royal Canal came through to link Dublin to the Shannon, with the similar Grand Canal further south. Like the rest of Ireland there was major loss of population in the wake of the 19th century famines, and from the slower but inexorable process of farm mechanisation. By 2016 the county population had rebuilt to 88,770, concentrated in the towns, with the countryside thinly populated. Water activities on the rivers and lakes are the main tourist attraction. There are TICs in Mullingar and Athlone, and the county agency is Visit Westmeath, which promotes the area as part of "Ireland's Hidden Heartlands".
Dublin Airport (IATA: DUB) is closest, and usually the best choice for its range of flights and good onward transport.
Trains from Dublin Connolly run via Drumcondra and Maynooth to Mullingar. They continue northwest to Longford, Dromod, Carrick-on-Shannon, Boyle, Ballymote and Sligo.
Trains from Dublin Heuston run via Portarlington, Tullamore and Clara to Athlone. They either continue west via Athenry to Galway, or northwest to Roscommon, Castlebar and Westport.
To Athlone, Bus Éireann and City Link compete on the route from Dublin Airport and city centre, taking 90 min (some stopping at Kilbeggan and Moate) and continuing west to Galway.
Athlone also has buses once or twice a day from Waterford via Kilkenny, Carlow, Port Laoise and Tullamore, and from Monaghan via Clones, Cavan Town, Longford and Ballymahon.
To Mullingar, quickest is Expressway Bus 22 / 23, taking 90 min from Dublin city and airport, Lucan and Maynooth. These continue to Longford whence Bus 23 runs to Dromod, Carrick-on-Shannon, Boyle and Sligo, while Bus 22 heads west to Ballina. There's also a slower Bus Éireann 115 from Dublin via Lucan, Maynooth, Kilcock, Enfield and Kinnegad to Mullingar.
By road follow N4 / M4 to Kinnegad, then take either N4 for Mullingar or M6 for Kilbeggan, Moate and Athlone.
There are two separate transport routes across the county, with poor interconnection: the bus between Mullingar and Athlone only runs twice a day. The two railway lines start from different Dublin stations so you don't even have the option of backtracking. The lakes and rural sites have next-to-no public transport. So you need wheels to get around: the terrain is lowland and a bike would do.