I'd advise two things. First, get new wheels. The Trek 1200 ($830; www.trekbikes.com) is a perfectly decent bike, but its wheels are not the best. Spend $225 U.S. (or thereabouts) and get a pair of wheels built on Shimano Ultegra hubs using Mavic Open Pro CD rims. Assuming the wheels are built correctly, that is an absolutely bombproof combination.
And certainly, switching to a fatter tire will help. In fact, very skinny tires (around 20 millimeters) may not roll as well as fatter ones because the tire deforms more on the surface. Materials, air pressure, and other factors also play a big part in how well a tire rolls. In the end, I'd go to maybe a 25-millimeter tire as opposed to the 28-millimeter width because this is more of a touring-type size and offers a good mix of rolling ability, comfort, and resistance against pinch-flats.