As I've spend my whole life here on the East Coast, I figured I'd chime in with a couple of my favorite here.
Probabaly my favorite, and the most comprehensive is the B&O Railroad Museum in Baltimore (
http://www.borail.org/ ) This museum has it all, and for a serious railfan, it can be a multi-day experience. Additionally, more off the beaten path is their annex in Ellicot City, located in what was the B&O's first terminus station, which itself has some unique history being located in the original station building. It also has a fairly accurate HO scale model of the B&O route from Baltimore to Ellicot City.
http://www.borail.org/Ellicott-City-Station.aspx
My second favorite that was always a fun childhood destination for me is the NC Transportation Museum (
http://www.nctrans.org/). Focusing not only on rail travel but as well boats, cars, and air, they have a good collection of equipment from a variety of short lines and little known southeast regional railroads, as well as equipment from some of the larger lines like Southern, SAL, and N&W. They even have an Amtrak F-40 that they purchased in the mid 2000s, however I haven't been back to see it on display.
A third favorite of mine is a little known museum in Chesapeake Beach, MD, the Chesapeake Beach Railroad Museum (
http://www.cbrm.org/). The building is the original terminus station for the Chesapeake Beach Railway, a former shortline between Washington, DC and the MD coast, and it has some interesting artifacts from the railroad including some CBR rolling stock.
Two other museums that are on my list of places to go but that I haven't yet been to are the Railroad Museum of Pennsylvania (
http://www.rrmuseumpa.org/) which I have heard excellent things about, and the C & O Railway Heritage Center (
http://www.candoheritage.org/), which I want to visit as I have always been fond of the C & O and have a family connection (among many) who was an engineer for that road in the 30s and 40s.
Great Idea for a topic, as I'm sure I'm missing several more mid-atlantic museums, and am hoping some other board members here will chime in and provide me suggestions for places to visit.
~ DCTE