As a vegetable gardener and a specific tast for Asian cuisine, one of the oldest and most refined cuisines in the world, I prefer my vegetables crispy and my meat to melt in my mouth. This is opposite of they way most average Americans were raised with tough, dry meat (like chicken breast) and oversteamed, mushy vegetables.
Try your vegetables fresh from your garden or farmers market. The produce will be better than grocery store produce. Then try sauteeing or steaming for only a couple of minutes. Literally (of course this depends on the vegetable). I especially prefer broccoli this way. The less they are cooked, the more body they have, crunch, and nutrients.
Try julienned jicima in salads.
Also consider roasting root vegetables. Many chefs consider this the best method by which to cook root vegetables for maximum complexity and depth of flavor.
Pick up a good cookbook and try a recipe or two a week. Vegetables
by James Peterson is a good place to start.