A regular sharpening stone is exactly what you don't want to use, because ski edges need a specific angle (typically between 88 and 90 degrees) and a normal stone will mess it up in five seconds. Plus, with that oxidation you mentioned, you first have to remove the corrosion without damaging the metal; it's not just sharpening, it's restoring.
If you really want to do it at home, you need a proper edge sharpener - those devices that come with fixed-angle guides and specialized stones. You can get them relatively cheap (like $30-50 or less) and they work pretty well if you use them patiently. The process would be: first remove the oxidation with a coarse-grit stone or even fine wet sandpaper, then move to the sharpening stone from the kit at the correct angle, and finally polish with a finer stone. The tricky part is keeping the angle consistent the whole time without screwing up; most people end up with one side sharper than the other.
That said, if your skis haven't been used in years and they're oxidized on top of that, it's probably worth taking them to a shop. It's not as expensive as it seems, and the guys there have machines that do the job with precision in twenty minutes. If you're going to use the skis regularly afterward, consider getting them tuned at a shop each season: it's basic maintenance. Saving a few bucks sharpening them yourself and then skiing on mediocre edges doesn't make much sense.