I go to school in Lancaster so i should be able to help with this one.
I know your not eager to use it but I really think the train is the best option. The Amtrak station in Lancaster is nice and easy to navigate. The Keystone train stops at EWR. From there they can take the free AirTrain to their terminal. The train station at EWR is only accesable from inside the airport and is extreamly safe. There are a bunch of people with red jackets that stand around the AirTrain station to tell people where to go. My only reservation about the train is baggage. Lancaster has a redcap (singular not plural) I don't think they have any rafications if they guy is out sick, day off or w/e. However people in Lancasater are very nice and I am sure someone would be happy to help. the real problem is at EWR. After you arrive at the EWR rail station you need to go up an esclator and walk a short distance to the AirTrain. Once again someone will probably volunteer to help but its a little less reliable since there is a lot more going on at EWR. If your bags have wheels and you can handle them on your own you'll be fine.
I take the Keystone a lot is it is almoast always ontime or within 10-15 minutes. The train is also comfortable.
Driving from Lanc to EWR is going to be a lot harder than than taking the train with parking, traffic, tolls ect. The only car/taxi service i know of in Lancaster is Friendly Transportation and they are horrable. I would never trust them to get me to/from EWR at any price.
Capitol Trailways is the only scheduled intra-city bus company that operates in Lancaster, PA (as far as i know) They do not go to EWR. You would have to switch to a greyhound bus to get to Newark and they a local bus to the airport. So I would rule this option out. Also an Amtrak train is 10x more comfortable than a bus.
If you have any more questions please let me know you can post them here or PM me. I am a Junior and travel a lot so I should be able to answer just about any question you have about getting places from Lancaster.