If I buy a solar panel that outputs 1.2Amps @ 15Volts (18watts) then a battery that requires 30 amp-Hours to be fully charged it would take 25hours to charge and with 2 panels it would take half as long.

It is posable to over charge a battery that is why it is important to buy a overcharger.

To increase the amp-hour out put of the system I need to add another battery. For example a 12Volt 30amp-hour outputs 30amp-hours to double this out put to 60 amp-hours I need to add another 12Volt 30amp-hour battery.

How to double to Voltage ouput or amperage output.
I have a 12Volt 30Amp battery.
I want to double the Voltage output to 24Volts
Connect them in a series
connect 2 of the above batterys like this; positive to negative positive to negative
I want to double the Amperage to 60 amps.
Connect them in parralel
connect 2 of the above batterys like this; positive to positive negative to negative.

If a 20 watt solar panel received 5 hours per day of full sun, then you would have generated 100 watt-hours of energy.
So if I was using a 20 watt compact flurescent bulb with the inverter (we'll call it 25 watts because the inverter uses some energy) then you could run the light bulb for about 4 hours with the energy you produced that day.
If I had a battery that holds 35amp-hours or over 400watt-hours of energy, then I could light the light for 12+ hours before I needed to recharge, but its better to charge sooner.

1.68 kWh per day to run my fridge