This list of specific vegetables will get you started.
Asparagus: Cut off ends. Soak in water for 30 minutes to an hour. Pat dry and brush with olive oil. Place on grill, turning every minute. Remove when tips start to turn brown. You can add some extra flavor to asparagus by mixing a little sesame oil in the olive oil before you brush them.
Bell Peppers: Cut through the middle of the pepper top to bottom. Remove stems, seeds and whitish ribs. Brush lightly with oil and grill for 2-3 minutes on each side. See Fire Roasted Peppers for more information.
Chili Peppers: Brush with oil. Grill whole on each side, 2-3 minutes. To reduce the heat, cut off the stems and pull out the seeds.
Corn on the cob: Gently pull back the husks but don't remove. Remove the silk and cut off the very end. Soak in cold water for about 30 minutes. Dry and brush with butter. Fold the husks back down and tie or twist the ends. Place on grill for about 5 to 7 minutes. Turn to avoid burning.
Eggplant: Cut lengthwise for smaller eggplants or in disks top to bottom for larger eggplants. Soak in water for 30 minutes. Pat dry, brush with oil and grill 2-3 minutes.
Garlic: Take whole bulbs and cut off the root end. Brush with olive oil and place cut side down over a hit fire. Grill garlic bulbs for about 10 minutes or until the skin is brown.
Mushrooms: Rinse off dirt and pat dry. Brush with oil and grill. 4-5 minutes for small mushrooms, 6-8 minutes. Use a grill basket or topper for small mushrooms.
Onions: Remove skin and cut horizontally about 1/2 inch thick. Brush with oil and grill 3-4 minutes.
Potatoes: There are many ways to grill potatoes. You can do them whole for a baked potato or cut into wedges, brush with olive oil, and grill until browned.
Tomatoes: (Yes, I know, technically a fruit) Cut in half, top to bottom. Brush with a light coating of oil and grill cut side down 2-3 minutes.
Potatoes: Wash thoroughly and dry. Rub with oil. Wrap in aluminum foil and grill 35-40 minutes, turning occasionally.
Zucchini and Small Squash: Slice 1/2 inch thickness. Brush with oil and grill 2-3 minutes per side. Small squash can be cut down the middle and grilling in halves.