I know that genetic diseases such as cystic fibrosis are often passed down through generations and are therefore classified as genetic disorders, but if a mutation occurs spontaneously, which for example leads to cancer, is this then classified as a genetic disorder?
Are all genetic disorders inherited?
Not all individuals with a genetic disorder inherited that disorder. Some genetic disorders are caused by spontaneous mutations.
Is cancer a genetic disorder?
Yes. The seminal paper by Hanahan and Weinberg, the Hallmarks of Cancer, is a good place to go to get a sense of what we understand cancer to be. This paper is quite influential and has its own wikipedia page. These authors wrote an updated review in 2011. Re: your question as it relates to cancer, yes, genetic changes result in tumorigenesis. Cancer is a genetic disorder of cell populations. One can view cancer diagnosis and treatment from the framework of cancer as a metabolic disease, but this adds to, rather than subtracts from the framework of cancer as a genetic disease.
Are other diseases caused by spontaneous mutations genetic disorders?
Yes. Achondrodysplasia is one illuminating example. It is a form of dwarfism caused by a mutation in the FGFR3 gene. It does follow mendelian autosomal dominant inheritance patterns, but in 80% of cases, the mutation is, in fact, acquired spontaneously (that is, not present in either parent). As noted by @Eff in the comments, Down syndrome, Trisomy 21 (and other chromosomal abnormalities, e.g., Turner, Kleinfelter, Patau, Edwards) are other common examples.