I try hard to get my kids to eat their vegetables - I want them to learn to choose to eat their vegetables, but also to get the important nutrition even when they don't want to eat them!
As mentioned previously, my second boy (D; 2 years old) is not a fan of most vegetables. However, we have had two successes this past week! D loves to watch, and participate, in meal preparation. This past week, I was preparing broccoli to puree for my baby (8 months) and had cooked fairly small pieces in boiling water until soft, then added it to some icy water (will explain). I was about to add it to the blender, when D indicated he wanted a bite. Not expecting him to do anything other than put it in his mouth, make a face, and immediately take it back out, I handed him a floret. He ate off the top (do any kids eat broccoli stalks?) and quickly did his "more" sign. I laid two more on the cutting board near him and he gobbled them down. This continued until I was out of broccoli! The boy ate about half of a small head of broccoli within 5 minutes! Other than an occasional bite or two of broccoli, enticed by "here comes the airplane/train/car" noises, he had never eaten broccoli in its true form before, so this was a big deal! I plan to try this method of cooking broccoli for him again (and for the baby, who did not yet get his first taste of this green veg).
The second success was a little more sneaky. I roasted red/orange/yellow peppers with a little oil to coat, salt, and pepper at about 400 F, along with some carrots and sweet potatoes. I pureed the peppers in my blender and then added them with some store-bought pasta sauce at about a 50:50 ratio and used this on homemade pizza and with noodles. Although boy #1 did notice (and actually did not care much for, but did eat), boy #2 gobbled it all down.
As you likely have noticed, the way that you cook a vegetable can really impact its texture and taste. Consider boiled carrots compared to roasted carrots - what a difference! Many kids enjoy raw vegetables with their crunchy textures, especially if there is a dip to use, but you should experiment with different cooking methods and seasonings. In general, the more fruits and vegetables (and most of us struggle with vegetables in particular) that are eaten, the better (although there are exceptions and limits). Across the world, many studies are done each year to see what kinds of foods promote health and which diets (meaning, the sum of what is eaten, not a "diet") are associated with negative health. Results do vary, but the one thing that is consistent is that more vegetables = better health. For more ideas of how to get kids to eat their fruits and vegetables, check out: www.eatright.org/food/planning-and-prep/snack-and-meal-ideas/get-fruits-and-veggies-to-the-plate .
*Adding the cooked broccoli to very cold water stops the cooking process from continuing. This helps to preserve the bright green color and the texture (so that it doesn't turn into a soggy mush).