When DD was a toddler I made her baby food. Gently steamed and then blended mild veggies with apple or pear. Over time, less of the fruit and more of the veg. Then just the veg and no blending.
Now it's a question of keep offering even if they go through phases of not liking it. You have to offer a kid something a lot of times to be sure.
I know from my spouse that texture matters. He'll eat raw apple all day long but not cooked ones. He objects to anything "squishy."
For child...Sweet ones like carrot, tomato, sugar snap peas, bell pepper or mild ones like green bean, zucchinni, romaine, cucumber, green pepper are always easy. So are "baby" veg. Well rinsed, drained beans like kidney or garbanzo are finger food snacks.
Bolder ones like broccoli or asparagus -- best to get it fresh as possible so it is still sweet. Then not to overcook.
It helps to let her make the salad or prep the veg where possible -- she's invested in it then and more likely to want it.
Also smoothies. Lots of ideas here:
http://www.amazon.com/Big-Book-Juice...3162662&sr=8-2
Last, she always has to take at least one bite and explain to be in detail what it is she objects to. The taste, the shape, how it is cooked etc. Then I'm willing to let it go and trade out that veggie for a small can of V8.
She just happens to like low sodium v8 -- I do not know why. So swapping out the veg for a v8 works when she tries something and it turns out not to work.
Sometimes I think the problem isn't so much the VEGGIE as it is the OTHER food. She doesn't get desserts at home or too much juice or sugary drinks. Fruit is about it. The sweetness in veg is often subtle -- and you aren't going to appreciate if getting a lot a sugar elsewhere. My own palate knows this... it isn't just kids.
She does like celestial seasonings lemon zinger, raspberry zinger, etc. I make it plain and iced so it looks like koolaid and pretty but isn't sugared up. We like the Honest Tea or Honest Kids pouches too because they are lower sugar in lunchboxes.
A.