For charcoal, may I recommend actual mesquite wood charcoal, instead of briquets. One brand is Lazzari, but you can find others. Mind, these aren't "flavoring chips", it's wood that's still recognizable but has been reduced to pure black 'coals'.
Why I like it:
First of all, I have to admit, it's fun. The stuff pops, crackles and sometimes puts on a show with a shower of sparks. Just be careful where you're grilling, and don't let hot cinders hit you.
Second, it burns hotter than regular charcoal. Seems to catch faster in a chimney, too.
Third, tasty smokey goodness.
Fourth, it burns completely leaving only fluffy white ash, not the clay-y binder materials which turn to cement if they get wet.
Fifth, it's all natural.
The downsides: well, it burns faster so you probably use it up faster. And it costs a bit more.
Cooking chicken on the grill is tricky but my wife likes to do it anyway; her method is to just cut slashes into the meat as it cooks, both to check for doneness and to expose more of the insides to the heat.
Two things to try in addition to the old standards:
1. King Oyster mushrooms (or other giant mushrooms). Slice them. Make mixture of melted butter (or olive oil), lemon juice (or balsamic vinegar), and some salt/pepper/herbs/spices. Brush it on the shrooms and grill.
2. Korean-style short ribs. Get ribs cut crosswise (like this
) and marinate them the night before in the stuff that comes in a jar like this
. You might have to try a few different brands at your Asian grocery store. Look for ones that have pear in the ingredients. Grill to taste, but I think it's best when the outer edges of the meat are a little burnt so that the sauce itself caramelizes.
Oh, and about "chimneys", you can make one out of a coffee tin supposedly, but I haven't tried. I got mine at the hardware store where they sell barbecue supplies.