Hi, E.! Well, there is no one answer for every age group and every situation. Nor is there one answer for every vegetable.
I am a true vegetable lover, so I actually have a hard time understanding why everyone doesn't love them. Some things that work well are making sure that the veggies aren't too bland. You can put some spice on them such as Mrs. Dash or other herb blends. Margarine or butter, of course, is good on things like green beans, squash, etc. If they don't like mushy things, try steaming the veggies instead. Try cutting them into different shapes. Try using cheese melted on top. Sometimes this helps kids eat things like broccolli. Cauliflower can be cooked soft and mashed like mashed potatoes. With a little butter, salt, and some spices, they might eat it if it is similar enough to mashed potatoes.
Alternatively, there is the meatloaf mystery mix. You can puree certain kinds of veggies and mix them well into the meatloaf, then serve it up with ketchup or whatever, and do not breathe a word of your secret invention. You can make smoothies with fruit & certain kinds of veggies -- sort of like V8 Fusion or whatever it's called, but watch out because some of those blends have artificial sweetner in them, and they can make some people sick (I'm desperately allergic to ALL artificial sweetners).
You can try them on things like carrot-orange juice or carrot-apple juice. Don't tell them that there's carrot juice in the stuff, and they won't know the difference. Give it to them when they're really thirsty so they won't be too concerned with what's in it, and then if you absolutely have to tell them, tell them after they've sucked down the entire glass. Yes, for stubborn kids who would rather starve than eat an unfashionable vegetabble, juices might be the best way to go.
Understand, though, that kids don't have to eat a whole lot of veggies all the time. They can get minerals from milk and fruits. They can get vitamins the same way.
Peace and good health,
Syl