Children like sweet food, rather than bitter tasting stuff, and they are conservative, meaning that they don't like to try new things; they have to see food a few times before they will try it according to some psychologists, they have a term for this- "neophobia"- the fear of new things. One way of overcoming this is to engage your child in making a menu for a week or a day or two at least, and taking them shopping, so that they can see what is displayed on shelves.
You can make a deal with them that they can choose vegetables, and fruit if this is also a problematic area, as long as they eat it. You can also allow children to "help" with the cooking and preparation of vegetables, or at least they can watch as you make a carrot flower or a tomato rose. These can be given to the child to try raw, but they do look enticing on a plate as nutritional garnishes. You can also deseed bell peppers and make them into containers for cream cheese or other dips that your child enjoys and gradually get him or her to eat the container. These can also be made into baskets to hold carrot and celery strips.
We eat with our eyes, so to get children to eat anything, apart from sweet things it has to look appetizing. You can make smiley faces on pizzas and quiches with different vegetables, red bell peppers make good lips and a carrot strip can be made to look like a nose, while the eyes can be green peas or strips of green pepper and so on. If it looks interesting it may be eaten.
Children don't like strong flavours, and if you recall your childhood you probably realize that you didn't approve of brussel sprouts or soggy boiled cabbage, or even asparagus which is considered a gourmet food. Go for caramelized onions and glazed carrots and cook your peas with mint sprigs to produce a sweeter taste.
Often it is the texture of vegetables that kids don't like, so puree veggies and take the seeds out of tomatoes and see if that helps. Make your own sauces for pasta and shred carrots into them and put in peeled, de-seeded tomatoes with onions and try a little garlic and oregano to help the taste.
Make your own soups and puree them so that they don't have to be chewed. If your child will eat mashed potato (perhaps smothered with tomato ketchup) then you could mash cooked carrots or swede with this and so disguise the fact that there are vegetables in it. You could add grated cheese too as this will help disguise the taste.