This depends on some factors such as quality and quantity of DNA extracted, primers, length of sequencing reads and existing of similar sequences in NCBI.
ITS primers are general and can be used for identification at genus and species level but at the species level, It may not be accurate.
At the first, you can use ITS primers and then if it does not work, you must use a specific primer.
The ITS region is generally a good marker for differentiating species. However, the intraspecific variation might not be enough and it is not phylogenetically conserved.
If you plan to do any phylogeny at family level as you mention, I wouldn't recommend it, at least not by itself. The D1-D2 or D1-D3 sections of the LSU have been used more and more in the last years for being more variable than the SSU, but unlike the ITS is more phylogenetically consistent. Otherwise, the EF or alpha-tubulin might do, but I have no experience with them so I can't tell you for sure.
The ITS region is a good marker to differenciate between fungal genera and species, however it is not enough for the intraspecific differentiation and can not be used in phylogenetic analyses.
There are definitely some species (eg in the Saccharomycotina) where ITS has a good intraspecific phylogenetic signal, but in general it is better for species determination (and in many genera it cannot distinguish every species). It all depends on the rate of speciation in the genus, the rate of concerted evolution, and the species definitions of the systematists who worked on that genus. In the Pezizomycotina, intron-containing sections of nuclear, protein-coding genes (eg actin, beta-tubulin, elongation factor) are preferred for intraspecific resolution (and species determination for rapidly evolving genera).