The key to getting good flavor out of a gas grill is very thorough preheating. You want your grill to reach temperatures upwards of 500-600 degrees Fahrenheit, at least. Your grates don't need to be glowing, per se, but they should be very, very hot.
If you're cooking burgers or steak, you shouldn't need to turn the heat down below medium. In fact, if you cook your burgers uncovered, then you can leave the temperature at the highest it'll go the whole time. This does use more gas, though, which is why most people put the meat on the very hot grate, then turn down the heat and close the cover.
Chicken, especially bone-in chicken1, is a different matter; you will have to turn down the heat, and/or use indirect heat (turn on a heating element on the opposite side from where your meat is), otherwise you'll end up with raw chicken coated in cinders.
1 Personally I don't cook bone-in chicken on the grill, because it's just so hard to get it cooked through properly, and there are so many better ways to cook it. (It's like, yeah, you can use your grill to cook pasta, but why would you? Unless of course there's a power outage or something...)
If you have a very low-end grill, it may not be capable of putting out enough heat to achieve the preheating temperature you want. Check out some grilling cookbooks, they will usually have guidelines on BTU's and such.