Answer is: no reaction occurs, because all salts are soluble in water.
Balanced chemical reaction:
Zn(NO₃)₂(aq) + MgSO₄(aq)→ Mg(NO₃)₂(aq) + ZnSO₄(aq).
Ionic reaction:
Zn²⁺(aq) + 2NO₃⁻(aq) + Mg²⁺(aq) + SO₄²⁻(aq) → Mg²⁺(aq) + 2NO₃⁻(aq) + Zn²⁺ + SO₄²⁻(aq).
This chemical reaction is double displacement reaction - cations and anions of the two reactants switch places and form two new compounds.