What they said, and nice rims. Welcome to the forum.

-Beware bike shop naysayers on the upgrade.
-Best to have a donor bike and swap the drivetrain.
-Best to go with rear hub that holds Shimano/SRAM 8/9/10.
-7sp RSX, you only have to swap freewheels and get a new chain, solid metal shifters.
-8sp RSX, 105, RX100, 600/Ultegra are the best bet, will require different rear wheel. (8sp DA only if you also get the RD)
-Sora is OK for a budget upgrade, and will shift fine, but you're already swapping wheels, so you are better off with a donor or used group.

-Cheapest upgrade:
--swap 6sp freewheel for 7sp. $15 new.
--pick up new or used 7-sp STI shifters, $80-$100
--pick up new chain, $15 new.
--used calipers with more clearance for commuter tires, $25
You'll be on the same bike, running 7 speeds, modern shifting, minimal fuss/expense.

Remember, you'll need STI housing, new shifter cables. And work on things to say when passing other cyclists.