You should definitely replace the cassette and chain.

The no-name dual pivot calipers on your Trek 1.2 would probably work much better with decent pads like FastJake recommended, but those brakes did get called out on at least one review for being sub-par. I couldn't fault you for replacing them if that's what you chose to do.

My overall recommendation is just a little bit different than Jake's: I'd replace the cassette & chain, shifter, and brakes -- choosing brake calipers to match your levers' pull ratio.

So then it's a matter of comparing costs for those parts versus the cost of a replacement groupset. It looks like buying just the necessary parts could save you about $100. On the other hand, going for a complete groupset will replace all wear items (shifters, chainrings, bottom bracket) and gives you an opportunity to upgrade to 10-speed or change your gearing if you'd like.

(All prices from Ribble unless noted otherwise)

INDIVIDUAL PARTS:
Shimano Sora 3500 right shifter $69.99 (Jenson)
Shimano Sora 3500 Brakes: $26.42
KMC x9.73 Chain: $10.56
Shimano HG400 9-speed 11-28 Cassette: $18.91
TOTAL: $125.88

COMPLETE GROUPSETS:
(levers, derailleurs, brakes, crankset, bottom bracket, cassette, & chain)
Shimano Sora R3000 9-speed Groupset: $222.58
Shimano Tiagra 4700 10-speed Groupset: $323.48 (Wiggle)
Campagnolo Veloce 10-speed Groupset: $333.87 (Wiggle)