Having asked a lot of questions about what to do because the Mary Gibbons Newgrange tours were booked up (reserved for Trinity College) 6 months in advance, I want to report back on our successful trip there by train and taxi.
I wavered as to whether to try to get to Newgrange because it would eat up an entire day of our Dublin stay. But then I realized that if I booked a late Newgrange tour spot we could do something in Dublin in the morning, take an early afternoon train to Drogeda, and still get back to Dublin by early evening (the fact that there's still daylight in Dublin at 10:00 pm is a big plus!). So we had a delightful breakfast at Lemon Jelly cafe, and then took the 10:00 a.m. Henrietta Street tour, which was excellent. We walked from Henrietta Street to the train station, stopping en route to pick up a sandwich, chips, etc. for a picnic lunch.
When you travel to/from Drogheda, look for the 37 minute express trains vs the more frequent hour long trains. We had an easy trip to Drogheda, and were able to pick up a taxi waiting at the train station. (There is also an option to travel to the Visitor Center by bus, but that takes over an hour. Our trip by taxi was about 20 minutes, and worth the extra cost in our opinion.) We got to the Center with time to spare, and enjoyed our lunch at one of the picnic tables there. We LOVED our visit -- really informative and unlike anything we'd ever seen before.
If possible, I'd suggest you try to plan to have the taxi that dropped you off return to pick you up at a specific time. We had planned to call our driver when we were done with our tour, but I was having phone problems, so the visitor center called a taxi for us. They have a list of taxis and are happy to make the call.
Here's what not to do: By the time we'd finished our visit and were waiting for the taxi, the center had closed and we needed to wait outside the gates. The person who called the cab told us it would be there in about 10 minutes, and sure enough about 10 minutes later a car pulled up. We asked "Drogheda Train Station?" and the driver confirmed. However, midway back to Drogheda we realized we didn't see a meter. When we asked the driver about that and what the ride would cost, he replied, "No English. Ukrainian." We were a little concerned but figured that if he did try to overcharge us we'd argue about it and show him our receipt from the trip to the center; if he tried to gouge us we could always report him after the fact to the Visitor Center.
We got into Drogheda and our driver pulled in to a Veterinary Clinic and stopped. That was weird. We told him we needed to go to the train station (we had reserved tickets and wanted to get on the express train). No English. "choo-choo!" He got out Google Translate and said "I need to wait for my daughter. She'll be here in 10 minutes." We responded "No, we specifically requested a taxi to the train station." He replied through Google translate "there's a bus station near here," and got out of the car to smoke a cigarette. FINALLY we realized what had happened. He wasn't a cab driver, just a nice guy who saw two people by the side of the road and pulled over to give us a ride to town! He had come to town to pick his daughter up from work. His thanks for being nice was being yelled at by two people who insisted he take us to the train station!
Once we figured this out, we tried to pay him but he refused to accept any money. We started to walk to the station (30 minutes away on foot, according to a woman we met en route) but found a cab at a gas station who took us the rest of the way. I feel really bad for the real cab driver who drove out to pick us up and found no one there! Lesson learned -- don't assume that because you called a cab the car that pulls over to pick you up actually IS a cab. (I know this was really dumb on our part and could have been dangerous, but in our case it was just a very funny trip story with a happy ending.)