Demonstrating decomposition of hydrogen peroxide using iron(III) nitrate catalyst
Detection of oxygen: Detection of $\ce{O2}$ by a glowing splint is a good way to detect the oxygen. Also you could capture the gas by a simple fixture e.g.  and demonstrate the volume change in the receiver. This way you can actually measure moles $\ce{O2}$ produced (by $PV=nRT$) then moles $\ce{H2O2}$ decomposed stoichiometrically by the formula you've written.

Choice of catalyst: Using $\ce{MnO2}$ would be better if you want to make sure you have a catalyst. $\ce{MnO2}$ will not be consumed during the decomposition; I'm not sure about iron(III) nitrate.

Or, detecting change in $\ce{Fe(NO3)3}$ concentration: Addition of a very small concentration of potassium thiocyanate, $\ce{KSCN}$ (say 1/100 of your $\ce{Fe^{+3}}$ concentration) will yield a deep red product, iron thiocyanate ($\ce{Fe(SCN)^{+2}}$). If you have access to a spectrometer you can measure absorbance of the initial solution's product and the product after decomposing $\ce{H2O2}$.

Prove water was produced: The best way I can think of is to measure the (subtle) change of density of your solution before and after decomposition. Using $\ce{MnO2}$ would make this easy because you can remove/filter it as a solid after decomposition and thus measure mass/volume of your solution before and after. $\ce{H2O2}$ and $\ce{H2O}$ have small albeit detectably different densities at RT. $\ce{H2O2}$ is more dense than water so your density should decrease.