I guess, I started when they were first eating solid foods. My mom told me that people make the mistake of giving their babies sweet fruits before introducing the flavor of vegetables.
My children's first solid was poi (Hawaiian taro) and then rice cereal. Then, we introduced blended vegetables like carrots, green beans, peas, squash, sweet potatoes. We didn't give the applesauce until they were well into the vegetables. We gave the sweet carrots before the squash though and when the kids tasted the squash, they shuddered!
Then, as they grew, we just served a lot of vegetables. We love them ourselves so the kids just thought it was what people ate. Once in a while, one of them would turn their noses at something. We didn't force it. We said, "More for us!" And made it desirable.
Kids' tastes change. If you don't fret, they'll eventually come around. My son didn't like cabbage but now loves it. My husband dislikes zucchini, but I adore it and my kids love it, too.
There is a book out that tells how to steam and puree vegetables and put them into meatloaf and sauces, etc.
I would stir-fry broccoli with a bit of teriyaki sauce. Ok, so there's sodium and sugar in it, but my kids would gobble it up! You can stir fry a lot of different veggies--broccoli, cabbage, carrots, celery--with a bit of teriyaki sauce and serve over rice or noodles for a great vegetarian dish.
The kids also loved to dip everything so I'd make vegetable sticks and serve with several dippers: Ranch dressing, marinara sauce, hummus, cheese dip.
Show them how to eat artichokes. They're fun to eat. If they're really little, they can pretend to be vegetable-eating animals. Dinosaurs? We watched a nature show about hippos and then tried to get our mouths to open wide and chew like hippos. Not great table manners at the time, but it was fun and they didn't realize that they were chewing green salad! My daughter became a carrot nut when she fell in love with bunnies and tried to eat like one, taking tiny nibbles.
I also showed them how put some mashed potatoes on a fork and then press it into a pile of peas to get them to stick. It was a hoot!
Also, if you put anything on a pizza, kids tend to eat it! They used to think all pizzas came with green peppers and mushrooms.
About table manners: I explained to them that there was a time for good table manners and a time when I would let them play (at lunchtime at home usually.) Their father didn't think it was a good idea to play at the table!
But they did eat their vegetables! And their table manners weren't so bad either!
