Great question! The answer depends a lot on what you play. Both techniques have their advantages, and we highly recommend that you learn both at least a little.
Playing with fingers is most common when playing a classical guitar with nylon strings. 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.
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 audio of a downpicked power chord with overdrive is what pretty much defines those styles. Strumming chords on a steel string guitar tends to be richer with a pick, and you get a louder volume in general. Downside 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.
The trick is to hold the pick between your thumb and index finger so that only the tip of the pick is visible. There are many different kinds of picks to suit different needs and styles. You should try out several to find the best match for your playing style. 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.