Hi, this is my first time going America. I will be staying lancaster PA for quite some time, would like to know what are the available transportation from columbia, Lancaster PA to Washington DC and New york City? And lastly how long will it take to reach the cities and the price for the transportation? Thanks a million.

It's your lucky day
Amtrak will get you from Lancaster PA "LNC" to Union Station in Washington DC" WAS " via a connection at the 30th St Station in Philadelphia

"PHL"
Once you reach either PHL or WAS then getting a train into NYC "NYP" is easy
The route between WAS and Boston "BOS " is Amtrak's most heavily traveled route and as such it gives you the greatest amount of trains to select from.
Amtrak offers something called a USA Rail Pass that you may find to be a good value.

http://tickets.amtrak.com/itd/amtrak/selectpass

Enjoy America