0-2v to 0-10v amplifier
If you choose too low valued resistors, they would draw too much current and dissipate too much, which isn't needed. (And when they draw really too much current, the may overdrive the OpAmps as well and/or damage it).
If you choose too high valued resistors, your circuit may be prone to external noise. And next to that, the OpAmp draws a small input current as well. With too big resistors, those currents influence the circuit too much.
Depending on the voltage of the OpAmps you pick the resistors.
I'd advise to start reading about RC filters and apply them.  
Leaving that to others. I'm not that experienced applying them.