Tiagra 9-speed stuff works well and is available in new condition. Why go any higher?

Your calipers are likely fine. Remove and clean them, and replace the pads with KoolStop dura salmon cartridge pads. That'd be the most effective braking upgrade.

You'll need a cable stop for the downtube, too.

So, that's shifters, cable stop, chain, cassette, cabling and brake pads. Should be well under $200.

Say again what's wrong with the 7-speed setup? Why not stay with 7-speed and go to the ST-A070 brifters? Cheaper...