As others have said, you'll end up wanting to get the fattest tires your bike can fit. Michelin used to make a tire called the Dynamic that's cheap ($12 or so each) and comes in a variety of sizes between 25 and 35mm. I had a pair of 32's on my old touring bike and loved them. They probably still make them.

But the real trick, with any tire, is to figure out what pressure to run. If you run any tire, fat or not, at "road bike pressures," (120psi+), you'll be pretty uncomfortable on less-than-perfect roads. Fat tires might help a little bit, but any increase in comfort will come from the slightly increased outer circumference of the wheel and any minor differences in the casing of the tire.

Lower pressure allows the tire to absorb more of the force of a given shock before it gets to your hands/feet/ass. You can probably manage to make your current bike more comfortable by lowering the tire pressure a little, but you'll probably be limited in how much you can do that by the fact that skinny tires can't deform much before pinchflatting. For reference, I weigh about 200lbs. On my road bike, I normally run about 120psi in each 23c tire. I can get down to about 100psi and still feel comfortable that I won't pinchflat. I can limp along with 80-90psi, but I have to be careful.

The real advantage of fat tires is not that they're inherently more comfortable; it's that you can run lower pressures without getting a pinchflat. Exactly how low depends on your weight, your bike's weight, and the roads you ride. Again, just for a ballpark reference, on my bike with 32mm tires I feel perfectly comfortable in the 70-90psi range. At those pressures, you barely feel cobblestones. It's heavenly.

Sheldon brown has a page on tires here, with a special section on width and pressure:
http://www.sheldonbrown.com/tires.html#pressure