Your digital piano is equipped with a damper, or "sustain" pedal. When depressed, notes will continue to ring until the pedal is released or they fade away on their own. Without the pedal, notes will only ring as long as you hold down the key.
If we had 88 fingers, we wouldn't need the sustain pedal because we could control each note's sustain individually with the key. [Yes, I know this is an oversimplification.] However, music is often written such that certain notes or passages (more than we have fingers for at one time) should ring out until we reach a point; usually a chord change.
The short answer is you should hold down the pedal whenever you want your notes to sustain, and release it whenever you want them to stop.
To learn when the notes SHOULD sustain, your suspicion about the Ped. and * symbols are correct: 'Ped.' signifies you should press and hold the pedal, and '*' signifies it should be released. You may also see an upwards-facing bracket underneath the bottom staff with upside-down 'V's along it, signifying that the pedal should be quickly released and repedaled at those points.
You will eventually use your musical intuition in addition to notation to figure this out. Take Debussy's Arabesque No. 1, for example. This piece is typically bathed in pedal, but the edition will generally not explicitly define it.
Here's a common usage:
Say you had some [bigchord]s, one after another, all different. You need to lift your hands from the keys of the first [bigchord] to set up for the next [bigchord], but doing so would cause the sound to stop while your hands are off the keys, and you want these [bigchord]s to connect (perhaps they are under a slur or legato 'arc'). You would depress the pedal in time to hit the first [bigchord], lift your hands to set up for the next [bigchord] (while the first is still ringing under the pedal), then release the pedal right as you play the next [bigchord], with your hands; repedaling before you release the hands from the keyboard to play the next [bigchord]. This sequence of resetting the pedal at the press of each chord is quite common.
On a real piano, each repetition of this would be lowering the dampers to the strings briefly, in effect "canceling" the set of harmonies that was ringing from the previous chord in time to play the next chord. If you had just held down the pedal through the entire sequence, each subsequent chord would add dissonance to the notes of the first chord that are still sustaining. Notes of different chords typically clash with one another in a typical tonal context.
Not playing staccato does not imply legato. Staccato simply asks for space by shortening the note in question. A note with neither an arc or dot is just a note of its full rhythmic value. Notes under an arc, legato, are specifically meant to be connected. In a single line, this may mean each note blends into the last one before it is released. With chords, this typically requires use of a pedal to sustain the sound before each articulation.
At the end of the day, listen to lots of classical piano recordings, experiment with your instrument until you have an understanding of how it works, and then do what sounds right.