If you have access to a centrifuge, either spin columns or phenol:chloroform will give very pure, high-quality DNA. Even with dirty methods, you can re-extract the product several times, to sacrifice yield for purity. Just be very clean. Whatever method you use should remove salts and proteins, which are the major contaminants. I won't go too much into how to extract DNA, as that's a whole separate question - but the most important thing is that purer is better.
You should resuspend in Tris-EDTA. Tris buffer maintains pH at slightly acidic (best for DNA) and EDTA chelates metal ions which can catalyze oxidative damage of DNA. Very pure distilled water works too, but if you get contaminants they will do more damage.
Afterwards you can store the DNA for quite some time provided it's frozen. At -20C you can store DNA for years. It may not be perfect quality when you pull it out a decade later, but most of it will be good, and I'm sure it will be okay if you use a good sequencing method (and I'm sure sequencing technology will advance more in 10 years than the degradation of your DNA). The good news is that you can simply store it in your freezer, and it will be okay. -20C particularly is not hugely important, you can probably get away with -10C or so - the crucial thing is for it to be frozen, not liquid (this vastly reduces diffusion).
If you want to do it like a pro, you would put it under -80C. This is considered very safe for multiple years. You can do even better by precipitating it with ethanol, and then storing at -80C. In my opinion, this -80C is overkill - DNA is fine, for the most part, stored at -20C and even 4C, especially if you work reasonably clean and contamination-free. If you had some extremely valuable, very sensitive DNA, sure, do the -80. But just personal sequencing/genotyping? Just extract a bit extra DNA (a couple micrograms).
The really safe method is liquid nitrogen, which stops practically all chemical activity. Higher temperatures are too cold for most DNA-damaging reactions to proceed at a significant rate, but the molecules still move around. At -200C, they stop dead. But, once again, overkill unless you are saving very critical DNA for centuries later.
Three simple things I can recommend that are less obvious: First, do not handle your sample with hands, even gloved. Use some kind of plastic tweezers. Hands are warm and will locally warm up the sample. I would go as far as to put the sample tube in another box so that there is no direct contact. Second, avoid freeze/thawing. Whatever your storage system is, make sure it is very reliable. Third, wrap it in something opaque, to minimize light exposure.
Note as per Chris's comments: Always consult the MSDS of the substances you work with and take appropriate safety precautions! Always make sure to receive proper training before working with hazardous materials, and never purchase illegal contraband without appropriate authorization!