When the grill is cool, remove the grates and flavorizer bars and put them into a full bucket of soapy water. Let them soak for at least 15 minutes. Remove anything else from the grill that you can take off without tools, like burner control knobs and the grease tray. This will make it easier to clean. Most gas grill burner tubes are removable as well (look for simple cotter pins on one end, although some models use screws).
Certain practices can help discourage deposits of dirt and grease from forming in the first place. One method is to grease the grates of your hot grill with a little cooking oil right before your start cooking. In the same vein, scrubbing your grill grates with a raw onion is another tactic you can try. If you do have a grill brush without bristles, it's a good idea to scrape your hot grates both before and after grilling.
It seems that people approach grilling steak one of two ways: either in a casual manner or paying perhaps too much attention. Some will heat the grill with abandon, throw the steaks on, flip once, then pull them off, and serve; others will fret and worry, cutting into the meat every two seconds wringing their hands over whether the meat is done or not. Both techniques do have some merits but a method somewhere in the middle is ideal.
Clean out the burners and venturi tubes. The venturi tubes are the pipes that go out from the burners and connect to the grill control valves. These tubes allow the air and gas to mix together, altering the intensity of the flame. Remove the burners and venturi tubes, and place a hose head at one end of the tube. Turn the water on to clear out any debris or insects that could have gotten inside.