You can not check your baggage on most trains that you travel on between Newark and (say) Washington, DC or Boston. But you could check your bags to Washington - they would just be carried on another train. That would mean you would have to check your bags in Newark before the earlier train departs or go back to the station in Washington to get you bags after the later train arrives.
Why is this? We are traveling Newark to DC and I would really not like to have to go back to DC to get my luggage. I guess I would rather know this now than be surprised.
The Northeast Regional trains that carry local traffic between New York and Washington do not have baggage cars, so they cannot handle checked luggage. (The exceptions are the overnight ones, trains #66/67.)
There are a number of long-distance services that originate at New York Penn and continue through Washington to points south and west, but for scheduling and revenue-enhancement purposes, Amtrak does not allow local traffic to ride these trains. However, most of these trains do have baggage cars and therefore can carry checked baggage.
So, if you check your bags in Newark, they will travel on the first available long-distance train offering checked baggage service. So, as the_traveler said, you can either drop your bags off before your trip (even a day or more early) and they will be waiting for you in WAS, or you can check your bags in Newark when you arrive at the station, and you will have to wait for them in WAS.
(If you can let us know what time your train departs or the train number, we can figure out how early you'd have to drop it off or how long you'd have to wait in WAS for it.)
A third option which was not mentioned is that you may not need to check your baggage. As long as you can easily handle your bags, there is plenty of room on-board to take a reasonable amount of luggage without checking them (the overhead racks are quite large--MUCH larger than even the largest airline overhead bins--and there may be an area for bags in the vestibule, although I don't know if the commuter-oriented Regional trains have these). As long as you can easily and quickly get your bags on-board and stowed, no one will challenge you. But if you struggle with your bags and can't move quickly, the car attendant or conductor may challenge you and not allow you to board the train with your luggage. So pack lightly!