You can use a gas grill for smoking and indirect cooking. Run the burner on one side of the grill, and place the food on the other. Wrap wood chips in foil, poke a few holes in it, and place the foil pack over the active burner. You can soak the chips if you like. I've never seen much point in that, myself. The water evaporates rather quickly, and you're back with dry chips. Better to limit the oxygen supply to the wood so it will smolder, and place the smoke pack somewhere so that it does not have too much heat applied to it.
It absolutely does matter what kind of wood you use. Avoid soft and sap-laden woods like pine. Stick with hardwoods or fruit woods. Hickory is often paired with pork. Apple and cherry are used ubiquitously. Oak is strong, but imparts an interesting flavor. Mesquite can be too powerful for some people -- I personally use it for grilling, but never for smoking.
Finally, be aware that you are going to have some limitations with your gas grill. It neither holds nor convects heat in the same manner that purpose-built smokers and some charcoal grills do. You are more likely to have hot and cold spots on your grill, so you will need to examine what you are cooking each hour and adjust accordingly if necessary. You will also not get nearly as much smoke to your food, so you will also have to adjust how much wood you use accordingly (though many people make the mistake of too much smoke in their barbecue, so a gas grill's limitation in this regard can be a benefit).