They are both really easy to get to. As Artem noted, you can take NJ Transit from Newark to NY Penn Station, which is more comfortable than the subway.
You can actually do an almost identical trip from JFK: you take the AirTrain from your terminal to the Jamaica stop, then you can transfer to any Penn-Station-bound Long Island Railroad (LIRR) train. You wind up in that same central, convenient Manhattan location as the NJ Transit from Newark option.
It may be a better option to fly into JFK, given that it's a weekend. It looks like the LIRR runs more frequently at that time of day on Saturdays (~3 times per hour). The NJ Transit trains seem to be at the :04 and :15 of the hour - so you could wind up waiting ~50 minutes for a train. (You can view the NJ Transit Schedule - select Newark Airport to New York Penn Station - and LIRR schedule - Jamaica to Penn Station - yourself.) Both of these options will cost you about $12-15 dollars per person one-way, and they drop you right into midtown Manhattan with easy connections to about 8 subway lines.
You also have the option of taking 2 buses from Newark to Penn Station, which Google Maps says is about the same amount of time (~40 minutes) as taking the train from JFK. That doesn't, of course, account for traffic you might experience on the bus.
JFK has the additional option of taking a subway into Manhattan. While cheaper ($5 for AirTrain + $2.25 for subway), I find that this option takes much longer, and if it's crowded, dealing with luggage can be a pain.
About.com has a nice summary of all of your different options for JFK-Manhattan travel, as well as EWR-Manhattan travel. (There are shuttle buses available from both airports to various locations in Manhattan, but I cannot speak from experience about them.)
My personal recommendation would be to fly into JFK; taking the AirTrain to LIRR is a nice middle-of-the-road choice that blends convenience, comfort, and affordability. (Taxis are expensive, buses are usually slower and more crowded, subways are hit or miss, and I've never taken the shuttle bus.)