I start off by asking my grandchildren questions, e.g., Do you like bananas? Do you like honey? Do you like brown bread? How about a little peanut butter? They answer yes.
Then I tell them we're going to make some very special sandwiches with all the things they like. The children love them because they were part of the process of trying something new.
Grandparents can sometimes introduce good old healthy recipes like they were brand new and exciting. With imagination you can make boring veggies look like an adventure or add their favorite cheese to a plain-looking dish. No, it doesn't always work...but just keep trying.
My 4-year-old fortunately LOVES veggies and fruit, as well as meat and potatoes (and, of course, sweets).
I let her fix her own salads, I set out the typical fixings, then include cherry or grape tomatoes, black beans, chunks of turkey or ham, cheese, red onions -- stuff that looks pretty to a kid.
I also let her make her own kebabs - while making sure she doesn't skewer herself! We use chicken, grape/cherry tomatoes, pineapple, green and red bell peppers, onions and then grill them. Again, these are very pretty and she enjoys getting to choose her own pattern for them.
My older girls loved "bug sandwiches" when they were small. I used a biscuit cutter to cut circles of bread, then made peanut butter & jelly or cheese sandwiches. Using small dabs of peanut butter, I stuck "bug stuff" on top of the sandwiches. Cheerios or fruit loops were eyes or spots. Pretzels or skinny slices of cheese were bug legs. You could also use raisins or other healthy stuff to decorate sandwiches.
I let my kids help with prep. Green beans and sweet corn are coming out of the garden right now, so I let the kids pick them. My 2.5-year-old can already husk corn and loves to help snap beans. It's hard for her to snap the little ends, but she has fun. She likes to eat them because she knows she helped make them.
We also can peaches. It's like an assembly line. First, the family picks them, always a lot of fun. The peaches have to be canned right away, so they don't get over-ripe. I par-boil them, and my oldest two kids skin and pit them. This year, the little one will help in some way -- maybe carrying them to and fro. All winter and into the spring, the kids have enjoyed eating "our" peaches.