We will be traveling to New York City in October. There is a flight into Newark NJ on Southwest; the hotel is near Times Square. Are we able to get into Manhattan from Newark International Airport (EWR)? Should we fly into LaGuardia in order to get from the airport to the hotel?
The distance between Newark and Times Square is longer than between LaGuardia Airport and Times Square, so based on distance, it may be better to fly into LaGuardia. However, it also depends on what time your flight is landing. If it's during the day (non-rush hour) then it probably won't matter which airport since it'll take you about 30 minutes to get to Times Square from each. If cost is a factor, taxi service from LaGuardia is cheaper than Newark.
As for your second question, you can get to a hotel via nearly all public and private transportation - depending on how well you know the bus/train system. You might be better off just calling a cab.
You can easily get from either EWR or LGA to Times Square, as there are several options to get you there. Most people make their decision based on which airport gets them the lowest fare, because of the plethora of ride options into the City.
From EWR you can take AirTrain/NJ Transit into Penn Station, from which Times Square is a short walk or cab ride - depends on how much luggage you're willing to carry half a mile. There are also buses and shuttles that can deposit you near Grand Central and/or Port Authority, which are also a short cab ride/walk to Times Square. Uber, Via, etc. also serve EWR - although these cost more than from LGA.
From LGA, there are shuttles into Manhattan, as well as taxis and other car services. If you are on a budget, you can also take the M60 SBS to the 2/3 at 125th Street, which you can take directly to Times Square/42nd Street.