NPN is generally better.
Just like the diagram above, you just connect the "ground side" of your load circuit to the NPN transistor's collector.
PNP high-side switching requires that the base be taken right up to the supply voltage (say 12V, or whatever) in order to turn it off - so if you've got a 0-3.3V microcontroller logic output you can't directly switch the transistor on and off without adding a second transistor.
Question - do you really need 20mA per LED? If they are ordinary LEDs, most modern LEDs are often quite bright with currents of only 5mA or so. Reducing the LED current will prolong battery life and reduce the size (current capacity) of the transistor needed.
As for the base resistor, to get technical about it, from first principles you need to know Ohm's law and the basic properties of a bipolar transistor:
VE = 0V
Therefore, VB = VBE, which is the drop across the transistor B-E junction.
IB = (VI - VB) / RB
IC = IB * hFE
RB = hFE * (VI - VBE) / IC
IC = say 400mA, the load current.
VBE = say 0.6V.
VI = 3.3V.
The typical hFE, the transistor's gain, can be looked up for your particular device.
Suppose the hFE = 100, then the above would give you 680 ohms.
As a "rule of thumb" I often skip the maths and usually just put a 1k resistor in any similar circuit.