The ITS is the genetic barcode region for fungi. Although it works fine for Basidiomycetes in some Ascomycetes it lacks resolution. 16s is for bacteria. 18s lacks variation in fungi.
An other option is the 28s (LSU) it is pretty conserved but allows you to allocate your samples in genera or families with confidence. It does not work good at the species level.
In conclusion: if you have time and money do LSU first and then ITS. If you want to save time and money use ITS but some samples will not have nothing close in public databases so wont be identified.