I personally cook on a natural gas grill. It costs a little more for the grill and to install the line but it's 100% worth it IMO. No filling propane tanks and no risk of running out of fuel in the middle of a cook out.
The type of fuel largely depends on what you're cooking IMO. Charcoal implants a smoky flavour to your food but that's not always desired. Like Zinfamous said, charcoal can also get really smoky too. I wouldn't want to use it if you have bitchy neighbours as it can really stink up the entire street. Charcoal is best for cooking blander meats like chicken, burgers, and white fish to give it a bit more flavours.
On the other hand, full flavoured foods or more delicate foods are best cooked on a propane or natural gas grill. Steak is at its best when it's cooked on a clean heat. You want the flavour of the meat itself to come through. All a good quality cut needs is salt and pepper. Same goes for pretty much any meat that has a higher fat content or stronger flavour. This includes beef and lamb. Vegetables too are best grilled on gas because you don't want them to have a smoky taste. Pork and veal can go either way. Veal is best cooked indoors though in sauces.
It's also easier to control the heat with gas. Certain types of food such as pulled pork or others that require slow cooking times or indirect heat will be much easier to cook. You can simply turn burners down or off. No need to push the coals around. Need more heat, it's on demand. If you miss the smoky flavour with gas alone, try adding some cumin or smoked Spanish paprika to your foods. I wouldn't use liquid smoke. . Gas grills are a good opportunity to play with different herbs and spices. Cumin is one of my favourites, along with strong herbs such as basil and rosemary.
When cooking on charcoal, I've been told lump charcoal is the best fuel as opposed to briquettes. I've used the latter and they're garbage. They take forever to heat up. The so called "easy light" ones are no better. Buy a
bbq chimney to preheat the charcoal. Makes it easier to start. If you want to get serious about grilling, it's a good idea to keep both types of grills on hand. Even a small portable charcoal grill is plenty to do up some burgers. You can get one for as little as $30. Larger ones are usually much cheaper than propane.