Currently it is possible and not so expensive for a person to have his genome. This is useful in general for understanding how life works. But, in practice, how is this useful for the specific person at this point in time?
A human genome sequence can uncover large deletions and insertions in the genome and would give the genotypes of both common and private (rare to very rare) small polymorphisms (e.g. SNPs) and SSRs (simple sequence repeats). From this information, one can learn about some curiosity traits (say, sensing asparagus metabolites in the urine, slow or fast twitch muscles, curly vs not curly hair) and, more importantly, about disease risk. Ancestry and family relationships can also be learned as in, for example, a half-sibling relationship.
As many diseases are polygenic with numerous loci each making small contributions to the variance, it becomes very difficult to fully describe the level of increased or decreased risk. In other words, we don't yet know all the players and so assessing their contribution to risk of disease can't be done with complete confidence. To further compound this problem are both epistasis and gene-environment interactions. Epistasis is a gene-gene (gene-gene-gene, etc) interaction, such that genes A and B separately make no contribution to the phenotype (disease risk), but do so in combination - say in ab/ab individuals. A gene-environment interaction can be described as when an allele associates with increased risk only when an environmental factor (e.g., exercise, fat in the diet, sun exposure, oxygen tension (altitude), etc), passes a certain threshold.
Still, the most powerful predictor of disease risk, onset and progression is family history. A genome sequence approaches that history (half your genome is from your mother and half from your father, of course), but remains fairly uninterpretable in terms of complex traits. You may carry a couple variants posing increased risk for heart disease, but what about the other hundred or so loci that also make small contributions to this affliction? Sure, if we knew exactly what those other hundred were and how much they each contribute and if they interact with each other or the environment. An incomplete picture emerges.
With rare diseases and altered genomes (say in cancer compared to normal) great strides are being made to either identify the small number of defective genes (rare diseases) or the commandeered pathways exploited for uncontrolled growth (cancer). In this regard, much can be learned and will have quicker, but still measurable in years, applications to the clinic.
How is the genome sequence useful right now? There are several tests out there that rely on identifying single mutations that will lead to diseases, or that are associated with bad response to medication. Having the entire genome sequenced allows identifying all of them at once, and is cheaper than doing these diagnostic tests one by one. Thus, sequencing the genome gives you information about whether to expect certain diseases, and a list of medication that you shouldn't take, sort of as an insurance.
Sequencing also allows you to learn about your ancestry, which may be of interest especially to people from countries with a long history of immigration, such as the US.
How is the genome sequence useful in the mid-term? Many diseases (including many cancers) are not caused by single mutations, but rather buy a combination of genetic and environmental risk factors that exceed some threshold. Thus, the genome sequence only allows to tell whether a person has increased risk for certain diseases, but not whether they'll get it. By itself, this information is not very useful. However, it has recently been suggested that blanket screening for cancer may not be the best idea due to bad side-effects and cost associated with treatment of false positives (which you'll get if you screen a sufficient number of people). Here, the genome sequence may help to identify who should get screened.
How is the genome sequence useful in the long term? The more genomes are sequenced, and the more phenotypes and diseases get associated with them, the better the predictions become, and the easier it will be for basic science to understand how mutations cause/influence disease. Thus, getting the genome sequenced now may be extremely useful and informative for the people who get their genome sequenced five years from now.
I assume some people would have their genome sequenced not so much for "My family has a history of heart disease - what risk factors have I inherited?" but for "I have my genome on this DVD - isn't that neat?".
The reason I suspect that is that I've heard that there are services that allow Europeans to find out which of the "Seven daughters of Eve" they're from, which seems to be more about curiosity than medical information.