Write the molecular, complete ionic, and net ionic equations.?

For the molecular equation you simply write down the molecules described above and you balance the equation.  All of them involve salt-metathesis (swapping of ions).  
The complete-ionic equation is where you write down all the ions and precipitate, gases, liquids, etc, present in solution.
The net-ionic equation is where you write down only the species that have changed chemically.

a) Molecular Eq:
ZnCl2(aq) + Na2CO3(aq) ---> ZnCO3(s) + 2 NaCl(aq)

Complete ionic:
Zn2+(aq) + 2 Cl-(aq) + 2 Na+(aq) + (CO3)2-(aq) ---> ZnCO3(s) + 2 Na+(aq) + Cl-(aq)

Net ionic Eq:
Zn2+(aq) + (CO3)2-(aq) ---> ZnCO3(s)

b) Molecular Eq:
LiNO3(aq) + MgSO4(aq) ---> no rxn

Complete ionic:
Li+(aq) + (NO3)-(aq) + Mg2+(aq) + (SO4)2-(aq) ---> Li+(aq) + (NO3)-(aq) + Mg2+(aq) + (SO4)2-(aq)

Net-ionic:

no rxn

However, if the concentrations are above 0.3 M then Li2SO4 will start precipitating and the net-ionic equation will be:
2 Li+(aq) + (SO4)2-(aq) ----> Li2SO4(s)

This will be specially true if the temperature of the solution is increased because the solubility of this compound decreases as temperature goes up (due to the negative entropy of reaction).

c) Molecular Eq:
2 (NH4)OH(aq) + NiCl2(aq) ----> Ni(OH)2(s) +  2 NH4Cl(aq)

Complete ionic eq:
2 (NH4)+(aq) + 2 OH-(aq) + Ni2+(aq) + 2 Cl-(aq) ---> Ni(OH)2(s) + 2 (NH4)+(aq) + 2 Cl-(aq)

Net-ionic eq:

Ni2+(aq) + 2 OH-(aq) ---> Ni(OH)2(s)

I hope this helped you understand the basics of this type of problem.