Coat your vegetables in a light oil before grilling them. It not only prevents them from sticking to the grill, it helps to prevent them from drying out. Don't be afraid to char your vegetables slightly - veggies such as corn on the cob, bell peppers, asparagus, zuchini, eggplant, and onions are terrific when you place them directly on the grill and let them get some nice dark lines across them.
Many people make the mistake of adding a sauce at the beginning of the grilling cycle. This can often lead to flare ups which burn the sugars and make a mess of the meal. Sauce them in the latter part of the cooking so the flavor comes through, not the burnt sugars.
It's often a good start to make a marinade that is slightly acidic. Start with something like vinegar, wine, citrus fruit, then a bit of oil and then add your spices. Try starting out with about 1 part acidic to 3 parts oil. The acidity of the marinade will break down the protein bonds in the food and tenderize it, while the oil will help to keep the food moist.
Don't marinade your food for too long - 30 minutes to 2 hours is a good rule to follow. Marinading for too long can make the food too soft, resulting in an unpleasant texture or even a tough, chewy texture - depending on the source of your marinade (enzymes from pineapple will make it mushy, while citrus fruits or vinegar will make them chewy).
Grilling with boneless meat is a different technique than bone-in meats. Boneless meats cook much faster on the grill and can handle the direct heat of the bottom grill. Meats that still have the bone in them take longer and grilling the on the direct heat of the bottom grill tends to burn the outsides will the inside is still uncooked. Use the top grill for the indirect heat of the BBQ for pieces with bone still in them.
Allowing grilled food to rest before service allows the juices in the food to redistribute evenly throughout.