Sound combination using op-amp
A summing opamp is the correct way to do this. However, your current configuration is problematic: you have the positive input connected to ground, as well as the negative supply of the opamp. This means that your signals need to be ground referenced (have a center point around 0v), but the opamp can't output voltages below 0v, so half your waveforms will be clipped off.

To fix this, either connect the opamp's negative supply to a negative voltage rail, or bias your signals to VCC/2, and connect the opamp's positive input to a resistor divider that also provides VCC/2.

Whether or not you need another amplifier before a speaker depends on the speaker; a small speaker is likely within the capabilities of your opamp, though it may not behave ideally. If in doubt, find a suitable audio amplifier IC and connect it on the output.