It could be argued that tires are the single largest contributing factor to vehicle performance. They're the only part of your truck that actually touches the road (or trail or rocks), and come in a wide enough variety that you could drastically alter the vehicle's performance without any other modifications at all. Tire size is so important that changing them is often one of the first things that performance enthusiasts do, followed immediately by suspension modifications to fit bigger ones.
Evaluate your needs. Big tires may look cool on the highway, but they'll do you little good if the truck rolls over or gets such bad fuel mileage that you can't drive it. As far as sizing goes, 285/70/R16 tires are about the largest you can fit on a Ram 1500, but expect them to rub when you turn the steering wheel at full lock; 265/75/R16 tires are the more hassle-free option. If you have a 2500-series, 33-inch-tall tires will fit with a minimal rub while turning.
Purchase the tires and have them installed at your local tire installer. You could do this yourself if you have the equipment, but you'll want to have the tires balanced anyway. You might as well just pay the additional $8 or so to have them mounted.
Test-drive the truck on a rutted road or trail to test for wheel rub. You're almost certain to experience a little bit under extreme circumstances; how much you consider acceptable is up to you. If the wheels regularly rub during regular driving on the street, you need to look into a 1- to 2-inch suspension lift kit. Lift blocks are inexpensive and easy to install, but using them on the front steer axle is dangerous. Opt for a set of aftermarket leaf or coil springs if you need to raise the front.
Correct your speedometer. Larger-diameter wheels travel farther per rotation than smaller ones, so your truck's speedometer can read several miles per hour under the speed you're actually traveling. Many Dodge trucks produced since OBD-II (on-board diagnostic, series 2) was introduced in 1996 have electronic speedometers that you can reprogram to compensate for the tire change. An aftermarket "power programmer" like those sold by Banks, Hypertech and Bully Dog will allow you to both correct the speedometer and re-tune the engine for some additional power. Otherwise, your local Dodge dealership should offer the service for a fee. Owners of older trucks will need to change out the speedometer cable gears in the transmission case for a new set calibrated to work with their tires.
