There are a lot of "correct" ways to cook vegetables but — though I'm not a huge fan of culinary presciptivism — I'm going to go ahead and say that roasting is the most correct. Everything from tender green asparagus to hearty root vegetables tastes phenomenal when prepared this way, and it's super easy to execute.
When preparing a 'square' meal - you know, the kind with a protein and at least two sides - I rarely struggle with the actual cooking. I can cook a chicken, mash some potatoes, and roast a pan of broccoli without any issue. But timing it all so everything ends up on the table simultaneously - hot and ready - is what gives me trouble.
The benefits of roasting vegetables are many. Not only is it a fairly hands-off method, but you don't even need a recipe or any additional ingredients besides a good cooking oil and some salt. Roasting also adds a savoury depth of flavour one can only achieve once the Maillard reaction and a bit of caramelisation have occurred, and crispy edges, which are a textural delight.
Not convinced? Just think about the differences between a boiled Brussels sprout and a roasted one. Where a boiled sprout is mushy, bland and offensively odorous, a roasted one is a slightly charred and sweet, with wonderful little crispy leaves that provide texture and a bit of bitterness. I actually can't think of a single vegetable that doesn't benefit from roasting. Even salad greens such as romaine can be roasted and transformed into more complex and intensely-flavoured versions of their fresh selves.
To roast your vegetables, simple cut them similarly-sized pieces and toss them with plenty of oil so they're completely coated. Lay them out on a baking sheet in a single layer, sprinkle on your salt and pepper, and cook them at 220C until they're fork tender and have crisped up at the edges. Timing-wise, the below should give you an idea of how long each plant part should veg out in the oven:
- Thin and soft vegetables: (Yellow squash, zucchini, peppers, green beans, asparagus, tomatoes) 10-20 minutes
- Greens: (Kale, mustard greens, collard greens) 6-10 minutes, depending on how crispy you want them
- Hearty crucifers: (Broccoli, cabbage, Brussels sprouts, cauliflower) 15-25 minutes
- Onions: 30-45 minutes
- Mushrooms: 20-40 minutes, depending on size
- Winter squash: 30-60 minutes, depending greatly on how small you dice 'em.
- Root vegetables: (Carrots, potatoes, sweet potatoes, beetroot) 35-60 minutes depending on size
Once you have a handle on roasting one vegetable at a time, you can start to pair similarly-timed produce, or add things in stages so everyone attains maximum roasty excellence at the same time. Also, know that it's OK if there are a few little burnt bits in the pan; burnt vegetables are pretty freaking good.