The burning of all fuels produce combustion by-products and smoke, which make food taste delicious, but are harmful to breathe. So regardless of which type of barbecue you choose, try not to breathe a lot of smoke.
There are basically two types of grills: gas and charcoal.
My husband and I barbecue over a small, inexpensive, portable charcoal grill. The point for us is to cook over the natural wood flame. We use a simple chimney-type starter rather than toxic lighter fluid, and we burn only natural wood briquets that have not been treated with any chemicals.
I'm not a fan of gas grills. They are more expensive to purchase, they require the purchase of propane gas for fuel, they are large in size, untilizing a lot of metal, which is very polluting to the environment, and in the end, it's not much different than cooking over a gas stove indoors.
You can read more about healthy barbecuing in my book Home Safe Home on page 309.