There are regular buses from Dublin to Drogheda. You can see the timetable here: http://www.buseireann.ie/site/home/. If you are arriving to Dublin airport you can go directly to Drogheda without going to Dublin city since at least some of the buses stop near Dublin airport on their way to Drogheda.
Bus or train is the best way to get to drogheda. Drogheda is about 30 miles from Dublin Airport and this is on the main Dublin-Drogheda bus route. The train station is about 15 minutes walk from the town centre. The bus station is near enough in the town centre.
The best way to get around Drogheda is on foot.
Throught the town goes the River Boyne, and over it is an old viaduct built which links the Dublin-Belfast railway line. It was the seventh bridge of its kind in the world when built in middle of 1800, and considered one of the wonders of the age. So taking the train is a good advice, or drive there by car.