Why do you cancel the ions out? Let's have a look.
If you react AgNO3 with BaCl2 in solution, the silver nitrate and barium chloride exist as solvated ions, not as ionic compounds. The only reaction that is really taking place is the precipitation of AgCl, so the only real reaction is Ag+(aq) + Cl-(aq) => AgCl(s)
Similarly in acid base reactions, if you react HCl with Na2CO3, the sodium ion doesn't do anything. Also HCl(aq) doesn't exist like that; it's actually H+(aq) and Cl-(aq), so the actual reaction taking place is H+ and CO3 2-, forming H2CO3, which decomposes to form CO2 and H2O.
However, you can only simplify things down if you have dissolved ionic substances or something is a strong acid/base. In the first example, iron(III) oxide is insoluble in water, but the HCl becomes H+ and Cl-, so the ionic equation would be Fe2O3 (s) + 6H+(aq) => 2Fe 3+(aq) + 3H2O(l)
Something similar happens with the second reaction. You can omit the sulfate in sulfuric acid.