I recently was given a propane grill that is a lot nicer than my charcoal grill, however my family really likes the cooking and taste of a charcoal grill. I hear the propane grill tastes a lot different. Is there something specific I can do to get a taste similar to a charcoal grill, while using the benefits of my newer, nicer, larger, propane grill?
Whether or not charcoal tastes different/better is actually a bit of a debate. I've been a propane griller for years and now I use charcoal not for the taste, but because frankly, I like to play with fire. But lets assume that there is a taste difference, identify where the differences comes from, and how to address them:
Those are the main tastes and how to replicate them. Nowadays the biggest is definitely #1, but it only matters if you doing foods that cook for a significant length of time or are especially 'porous'. If you're doing hot dogs and chicken breasts, I wouldn't bother if smoke. If you're doing burgers - maybe/probably. If you're doing thick steaks, thick pork chops, a whole chicken, or anything that's going on awhile, then creating some smoke is definitely worth it. ("Liquid smoke" definitely doesn't taste like the real thing.)
While I like the "Smoke Box" approach that @rfusca offers you can accomplish the same effect with an cheap/disposable aluminum tray (like those pot pies come in) and a few wood chips, it really doesn't take much smoke to get the flavor into your meats. Just place the tray on the grill slightly away from the meats and let it get to smoking before you put you steaks/burgers on, and keep the lid shut.
Another option is to use "Liquid Smoke". This must be done carefully and in small doses as a little goes a long way. You will get the best results out of mixing Liquid Smoke with something. For steaks I like to mix about a teaspoon with a stick of softened butter (along with garlic, rosemary, oregano if you like...) whip it up and then chill it. While grilling, after you have turned the steak add about a tablespoon of the butter. As it melts it will impart a subtle smokey goodness to your steak. (also works well with chicken)
For burgers mixing a tablespoon of the butter mixture in to the patty when you are forming it works well. However, I like to mix a little Liquid Smoke with some BBQ sauce and work that into the raw ground beef and then form that to patties and grill normally.
On an episode of Good Eats Alton Brown demonstrated how to make liquid smoke. Something which every serious griller should try, ONCE. ;)