Getting to Newark is easy by train since it is along the main line between London and the North of England. Not all main line trains stop here, but enough of them for you to be spontaneous. They stop at Newark North Gate which has a top part of the station along the main line, and a crossing line to for instance Lincoln a bit further down some stairs. Give yourself plenty of time if you want to change here and have problems with mobility, or lots of luggage.
Even more time is needed if you want to change between Newark North Gate and little Newark Castle which is the other station in town. It is a 15 minute (nice) walk, or a short taxi ride between the two, and Newark Castle is also used for trains to Lincoln and Leicestershire so you need to be careful when checking your travel details so that you get the right place. Newark Castle is, as its name suggests, close to the castle but also a supermarket and the town centre if you have some time to kill. Newark North Gate is more on the outskirts of the centre.
2 Rail stations
My Grandfather was (I think I am right in saying this) a signalman in Newark during the Second world war. As such he no doubt had a very responsible job to manage the main line connecting Edinburgh to London and the local lines that stop closer to town at the 'Castle Station'.
Newark Northgate is on the main line, and that means there are loads of high-speed trains available in both directrions. The station is however a rather annoying 15 minute walk from the town centre. The 'Castle' Station, just the other side of the river to the town deals with more local traffic - mainly from Nottingham. It is unlikely that you will need to interconnect between the two.