Balancing Ionic and Net Ionic equations

I am currently taking a chemistry course about acids and bases. One of the things I am a bit confused about, is the acid and base neutralization reaction, and how to balance it. For Example:

$$\ce{NaOH + HCl -> NaCl + H2O}$$ How do you write the ionic and net ionic equations for that?

Note: My main problem here is that my teacher requires us to first write the ionization of the $\ce{HCl}$ into $\ce{H3O+}$ and $\ce{Cl-}$. This is when I get confused, because I don't know how to balance the equations with the $\ce{H3O+}$

Also, I had another question in my exam, which I got wrong because of the balancing. Can you please tell me the correction of it:

Consider the neutralization reaction between nitric acid and calcium hydroxide. Write the balanced neutralization reaction and the ionic and net ionic reactions of it:

As $\ce{NaOH}$ and $\ce{HCl}$ are strong base and acid respectively, they are completely dissociated in water. The salt $\ce{NaCl}$ is a strong electrolyte and completely dissociated in water. So, the ionic equation is:

$\ce{(Na^+ +OH^- )}+\ce{(H^+ +Cl^- )}\rightarrow \ce{(Na^+ +Cl^- )}+\ce{H2O}$

As the ions $\ce{Na^+}$ and$\ce{Cl^-}$ do not combine and remain in the solution as spectator ions. Their role is to keep the solution electrically neutral, the net ionic equation is: $\ce{OH^- }+\ce{H^+ }\rightarrow \ce{H2O}$.

On the other hand, we have to notice that the ion $\ce{H^+}$ is not stable in water, it combines with $\ce{H2O}$ to form the ion hydronium $\ce{H_3O^+}$ . We usually write the complete dissociation of $\ce{HCl }$ as follows:

$\ce{HCl} +\ce{H_2O }\rightarrow$ $\ce{Cl^-} +\ce{H_3O^+}$

and the ionic equation is

$\ce{(Na^+ +OH^- )}+( \ce{H_3O^+}$$+\ce{Cl^- })\rightarrow \ce{(Na^+ +Cl^- )}+2\ce{H2O} And the net ionic equation is \ce{OH^- }+\ce{H_3O^+ }\rightarrow 2\ce{H2O} . As for your exam questions: 2\ce{HNO_3} + \ce{Ca(OH)_2 }\rightarrow \ce{Ca(NO_3)_2} + 2\ce{H_2O} Now you can try to write the ionic and the net ionic equation to see if you understand what I have explained. Consider the neutralization reaction between nitric acid and calcium hydroxide. Write the balanced neutralization reaction and the ionic and net ionic reactions of it For simplicity consider this:$$\ce{HNO3}\tag{nitric acid, ionic}\ce{Ca(OH)2}\tag{calcium hydroxide, ionic}$$So, they'll break into their constituent ions:$$\ce{HNO3<=>H+ +NO3-}\\\ce{Ca(OH)2<=>Ca^2+ +OH-}$$Now \small\ce{H+} exists as \small\ce{H3O+} and after balancing by adding water to both sides and doing \small\ce{H2O +H+\sim H3O+}.Now multiply the first reaction by 2 after we notice the two hydroxide ions in the decomposition of calcium hydroxide, since we want in an integral multiple, the reaction \small\ce{H+ +OH- ->H2O\equiv H3O+ + OH- ->2H2O}$$\begin{array}{}\ce{&2HNO3 +2H2O&<=>&2NO3-&+&2H3O+}\\\ce{&Ca(OH)2&<=>&Ca^2+ &+ &2OH-}\\\hline\ce{&2HNO3 +Ca(OH)2&->&\underbrace{Ca(NO3)2}_{Ca^2+ + NO3- <=> Ca(NO3)2}&+}&\underbrace{\ce{2H2O}}_{4-2=2}\end{array}$\$