Originally Posted by Wilfred Laurier

The only 'upgrade' that is likely to make any noticeable difference is the tires and wheels. Tires will wear out after a season or two of riding, so you can reward yourself for riding a lot with some better quality, faster rolling tires.
Wheels are a more expensive upgrade, but also will make a difference to the feel and performance of the bike. However, at 265 lbs, the machine built wheels may have a finite life (spokes likely to begin breaking at some point) so wait until they need to be replaced before you spend any money on them. It is likely the rear wheel, which holds most of the load, will need replacing before the front.

If you are doing proper maintenance on your bike - keeping the drivetrain clean and properly lubricated, keeping serviceable bearings greased and properly adjusted, and ensuring shifting and braking mechanisms are working properly with the cables and housing in good condition - then nothing aside from tires and wheels will make a difference. You might save a pound or two by spending a few hundred dollars on lighter handlebars or cranks or whatever, but this means almost nothing compared to your almost-300-lb loaded-bike-plus-rider rolling weight.

Save your money and keep your eyes peeled for a good second hand 'road' bike,. which will be lighter and more aero from the start. That will also leave you with this bike as a commuter or 'rain' bike for rides when you don't want to use the road bike.