Stepwise movement is called “conjunct” movement and is the most common type of movement found in all styles of music. Conjunct movement is the easiest to sing, and usually the easiest to play on most instruments too. You can find plenty of examples in any music you find, but here are a few illustrations.
In the children’s nursery rhyme Row, Row, Row Your Boat, the melody moves by conjunct motion throughout the first line (apart from the repeated notes):
Beethoven’s Ode to Joy (from the 9th Symphony) also begins with conjunct motion:
Movement by a leap of a third or more is called “disjunct” movement. Most tunes are built from a lot of conjunct motion, and a smaller amount of disjunct motion, and this is particularly true of vocal music.
The song Lavenders Blue is a good example. There are a few leaps in the tune, but most of the movement is by step.
After a melody moves by a leap, it will then most often move in the opposite direction, or, if it continues in the same direction, the notes will all normally belong to the same chord.
This is the song Star-Spangled Banner. The first two bars all move by disjunct motion in the same direction upwards, but all of the notes belong to the same tonic chord.
In bar 3, there is a leap of a 6th downwards, but this is followed by a turn in the opposite direction, to F#, because the harmony has changed here as well. The C and E belong to the same chord, but the F# does not. If the harmony changes, it’s best to change direction as well.