Vesuvan Shapeshifter copies exactly what is printed on the creature it's copying and nothing else (unless it's copying a creature that's copying something else or that's a token; see below). It doesn't copy whether a creature is tapped or untapped, whether it has any counters on it or Auras and/or Equipment attached to it, or any non-copy effects that changed its power, toughness, types, color, and so on. Most notably, if it copies a creature that's not normally a creature, it won't be a creature.

If a copied creature has {X} in its mana cost, X is 0.

If a copied creature is copying something else, Vesuvan Shapeshifter uses the copiable values of that creature. In most cases, Vesuvan Shapeshifter will copy whatever that creature is copying.

If a copied creature is a token that isn't a copy of something else, Vesuvan Shapeshifter copies the original characteristics of that token as stated by the effect that created it. Vesuvan Shapeshifter isn't a token in this case.

Any enters-the-battlefield abilities of the copied creature will trigger if Vesuvan Shapeshifter is entering the battlefield. Any "As [this creature] enters the battlefield" or "[This creature] enters the battlefield with" abilities of the copied creature will also work. If a Vesuvan Shapeshifter becomes a copy of a creature after it's already on the battlefield, these abilities won't apply as Vesuvan Shapeshifter becomes a copy since it's not entering the battlefield; instead, any "As [this creature] is turned face up" or "When [this creature] is turned face up" abilities will apply.

If something copies Vesuvan Shapeshifter, the new copy will also have the triggered ability to turn it face down. However, once face-down, it won't have Vesuvan Shapeshifter's morph ability, so it can't be turned face up unless whatever it's copying also has a morph ability.