Help with designing a circuit with TEC3-2419 DC-DC Converter
Why do I have Vout(+) and Vout(-)? From what I know about buck
  converters there are multiple switching pins, okay, but why both
  positive and negative Vout pins?


It's an isolating DC to DC converter and therefore has two pins for +Vout and -Vout. Those pins are isolated from other pins. It isn't a buck converter.


  Do I need an inductor at the output? If so, how do I calculate the
  value?


No, because it's an isolating DC to DC converter. It uses an internal flyback transformer for passing magnetic energy across an isolated internal physical gap.


  I suppose it would be a good idea to connect bypass capacitors at the
  input and at the output. How many do I need and what should be their
  values?


It has inbuilt input and output capacitors. Addding more at the input may reduce conducted emissions to other equipment sharing the same input DC system. The data sheet tells you how much you can add to the output (see page 3 of DS).


  Pin #3 is On/off. Does that mean EN pin? If so, should I put a
  resistor between this pin and the VCC pin (again, something I saw in
  previous buck converters). What value should the resistor be?


Page 3 of the DS tells you - open circuit means on and off is typically using a 1 kohm pull-down.


  What is the difference between single and dual as noted on page 4 of
  the datasheet? Or, is there a website you can direct me to? I have
  Googled it but couldn't find any information about this


Some of the models in the product range have dual outputs outputs i.e. a mid-rail common 0 volts and positive and negative output voltages. They are referred to as "dual output" on page 3.