Fingers or pick?
Great question! It depends a lot on what you play. There's no one answer though, you have to experiment and find your own style!
When should you use your fingers / a pick
Both techniques have their advantages, and we highly recommend you to learn both at least a little.
Playing with fingers is most common when playing a classical guitar with nylon strings, but it's not unusual to play fingerstyle on steel string guitars and electric guitars too. With fingers you can get a soft and rich tone, but also a sharp attack, e.g. when playing double stops. It's common to play the bass strings with the thumb and use index, middle and ring fingers for the treble strings. The legendary jazz guitarist Wes Montgomery used to play using only his thumb.
Some other great guitarists playing fingerstyle are Tommy Emmanuel, Mark Knopfler and Petteri Sariola.
Picking = using a pick / plectrum
Playing with a pick (or plectrum) is especially useful when playing lots of notes fast. Rock and metal riffs and solos are usually played using a pick. The sound of a downpicked power chord with overdrive is what pretty much defines the whole style. Also, strumming chords on a steel string guitar tends to sound better with a pick and you get a louder sound in general. Downsides to using a pick is that string skipping (jumping over one or more strings) can be more difficult than when playing with fingers, e.g. when playing arpeggio patterns.
Hold the pick between your thumb and index finger so that only the tip of the pick is visible. There are many different kind of picks to suit different needs and styles. You should try many to find the optimal match for your playing. If you play a nylon string guitar with a pick be careful not to scratch the surface, as they rarely have pick guards to protect the body.
There is also a technique called hybrid picking, that combines the use of a pick and fingers.
See for example Mr Fastfinger demonstrate this on his track "Practice Heart": Video