What is the way to cook vegetables that keeps most of the nutrients?
I prefer the microwave oven - you can cook vegetables in the quickest time in the minimum amount of water.
More taste even than steamed veg!
It depends upon the vegetable in question, the nutrients one desires to obtain from it, and the digestive capacity of the person (or, if you're considering cooking for your pets or livestock, animal) eating the result.
Take carrots, for instance. As this article discusses, "boiling and steaming better preserves antioxidants, particularly carotenoids, in carrots, than frying, though boiling was deemed the best." Also, "cooking carrots increases their level of beta-carotene." Yet, "The downside of cooking vegetables is that it can destroy some of the vitamin C in them. The reason is that Vitamin C, which is highly unstable, is easily degraded through oxidation, exposure to heat (it can increase the rate at which vitamin C reacts with oxygen in the air) and through cooking in water (it dissolves in water)."
Generally speaking, steaming is best as the nutrients largely remain in the vegetables. Boiling them leaches the nutrients out into the water.
Of course, for maximum nutrition, don't cook them at all!
If you're sweating about losing some nutrients, consider eating more of the vegetables in question to make up for this supposed deficit.
Nobody ever suffered malnutrition because they cooked their veg instead of eating them raw.