99% of bass lessons (this one included) start with fingerstyle plucking. But a pick is a legitimate technique with its own sound. Many great bassists use one (Paul McCartney, Mike Dirnt of Green Day, Chris Squire). Time to learn both.
How to hold a bass pick
Same as guitar: pick held between thumb and side of index finger. The pick should stick out about 3-5 mm. A heavier pick (1.0-1.5 mm) is standard for bass; thin guitar picks bend too much against bass strings.
Pick technique
Down-up-down-up alternating. Same as alternate picking on guitar. The pick rests against the string after each stroke; don't lift it all the way off.
For a quarter-note groove: all down strokes.
For an eighth-note groove: down-up-down-up.
Sonic differences
Fingers: warmer, rounder tone. Less attack. Better for funk, jazz, R&B, classic rock, anything with a "feel" emphasis.
Pick: brighter, more attack. The note has a click at the start. Better for rock, punk, metal, anything where the bass needs to cut through a dense mix.
Some songs demand a pick (you can hear it). Some demand fingers. Most can use either, and the choice is your sonic signature.
100 BPM. Play 8 quarter notes on the A string with a pick (all down strokes). Then play the same 8 with fingers (alternating I-M). Listen to the difference.
Try both for one week
Spend a week playing fingerstyle (which you've been doing). Then a week with a pick. Then pick whichever you prefer for now. You can switch later, plenty of pros use both depending on the song.
Next: your first real bass line, the Seven Nation Army.