First of all, the title says "Encourage" not "make them". So while your intentions are good in starting this vegetable diet for the kids, please, do not, in any way, make it traumatic for the children.
I am proud to have been raised in a family where vegetables are a regular fare and are even sought after and called comfort food. I come from an ethnicity called "Ilocano" and we are known for being frugal and for being vegetable eaters. We do love meat as well but, vegetables, they are important to us.
Now that I have my own kids, I feel it is necessary to get them on the same diet as well. I have long started on getting them used to the taste of vegetables. On a regular basis, we have fruits and vegetables in our meals. From soup to really all-out vegetable dishes like chopsuey, pinakbet or sinigang. Some of my kids even eat ampalaya or bitter melon or bitter gourd. I admit that there are days when they test my patience and refuse to eat. It is important for you to put your foot down on this and teach them to eat what is served for the meal.
What vegetables do they eat now? Eggplants, string beans, carrots, bottle gourd, chayote, squash, green beans, pechay, sweet potato tops, kangkong or swamp cabbage tops, green peas and more. So I am pretty happy with their diet.
Hope these tips help you.
# 1 Be a great example. Kids have to see you eat vegetables and actually enjoy eating vegetables. As simple as that.
# 2 Start simple. When others write about being creative, I feel like that is a sissy way. You cannot make smiley face out of carrots and cucumbers all the time. Also, for me, that is a coward way to teach your children to eat vegetables. Vegetables are not served in a shape of a nose or eye all the time!
Start with cream of squash soup, tinola (chicken soup with papaya or chayote and moringa or pepper leaves) or fried eggplant. Something that will not give them the ewwws or smirks. Accompany the food with something they like. On top of those, make sure that the food tastes good and that it is cooked in a healthy way. Stir away from spicy or bitter flavors initially.
# 3 Be consistent. If children see that there are vegetables everyday on the menu and that you eat them yourself, it will be easier for them to realize that veggies are going to be always on the table. Remember also to NOT serve anything else. If you give in and serve the dish that they like at attempts not to eat, there will be no victory for you. It's a discipline. If kids see that either they eat what is served or they starve, they will get used to it. Kids will not get sick or die if they miss one meal or cry. I will admit that for children 4 years old and younger, it might be more difficult so my advice would be to, again, keep the dish simple.
Remember also that the reason why you are doing this is for their health. If you are going to fry everything, that is not healthy as well. Broths and stews are a favorite in our house. We limit the fried stuff to weekends. Remember also that fruits are great in encouraging children to eat vegetables.
Start them early and hopefully, they will look forward to eating vegetables all their life.