Sorry I didn't understand you wanted to learn op-amp use.
I you just want typical bathroom scale accuracy the Arduino is fine. You won't need much amplification to get that level of accuracy.
The sensor in the scale is probably a half bridge like this Load Sensor - 50kg (Generic) - SEN-10245 - SparkFun Electronics.
Sparkfun has this link NerdKits - Digital Scale Strain Gauge Weight Sensor.
The video uses a full bridge sensor but you just need two resistors and/or a pot to complete the bridge. The video uses a true instrumentation amplifier, an AD620, which is expensive but easy to use.
18 - 24 bit delta-sigma ADCs are a better fit for this type application. A delta-sigma ADC requires less signal conditioning circuitry, less amplification, and has greater resolution than the Arduino SAR ADC.
There are lots of app notes like this on the web: