If ions are floating around in a solution, how do we know they combine to form a compound in a solution?
User EdV comments above get to the heart of the matter. To summarize:

Your first equation is typically how a chemist would write the reaction. 

$$\ce{3Pb(NO3)2 (aq) + 2(NH4)3PO4(aq) -&gt; Pb3(PO4)2(s) + 6NH4NO3(aq)}$$

The equation is a "shorthand" explanation. What a chemist reads is:


On the left hand side is that that two solutions are mixed together. 


Since $\ce{Pb(NO3)2}$ has an "(aq)" modifier it will be in solution, but since it is an ionic compound it will not be in solution as a molecule, but rather as the $\ce{Pb^{2+}}$ and $\ce{NO3-}$ ions.
Since $\ce{(NH4)3PO4}$ has an "(aq)" modifier it will be in solution, but since it is an ionic compound it will not be in solution as a molecule, but rather as the $\ce{NH4+}$ and $\ce{PO4^{3-}}$ ions.
Since the RHS shows a reaction the two salts can't be in one solution simultaneously. Therefore the salts must have been dissolved in separate solutions and the solutions were mixed. 

On the right hand side $\ce{Pb3(PO4)2}$ precipitates and $\ce{6NH4NO3}$ remains in solution. 


Also since $\ce{Pb3(PO4)2}$ has a tiny but finite solubility, a very small amount of $\ce{Pb3(PO4)2}$ will remain in solution. 
Since $\ce{NH4NO3}$ has an "(aq)" modifier it will be in solution, but since it is an ionic compound it will not be in solution as a molecule, but rather as the $\ce{NH4+}$ and $\ce{NO3-}$ ions.



The "ionic equation" is harder to understand because of its verbosity. Did the $\ce{NO3-}$ ions come into the solution with the $\ce{Pb^{2+}}$ ions or the $\ce{NH4+}$ ions? Being chemists we can puzzle it out, but the equation is less succinct. 
$$\ce{3Pb2+(aq) + 6NO3-(aq) +6NH4+(aq) + 2PO4^3-(aq) -&gt; Pb3(PO3)2(s) + 6NH4+(aq) + 6NO3-(aq)}$$