Most of an electricity bill can be due to hot water, and this can easily be replaced by sunlight.

This is a comparison of the different ways of heating water

Modern hot water systems

Electric heat pump - a vapour is compressed making it hot enough to heat water.

Solar hot water - close coupled. Tank is above the collector plates allowing a thermo-syphon to circulate the water without pumping. 

Solar hot water split system - Tank is on the ground and a small pump circulates the water. A little more expensive, but usually chosen for looks. It is probably more efficient because the faster flow allows more heat transfer.

Size

If you choose a flat plate area of 2 sq metres per person then you will have enough capacity to carry you over 2 or 3 days of cloudy weather.

Ref: Plumbers training handbook (No air temperature stated so would need to be calculated for each location.)

Close coupled, or thermo-siphon, requires no pump. The water heats under glass, rises into the top of the tank and sends more cold water from the bottom of the tank to the bottom of the glass plate.   

   Split system is lighter and neater on the roof, but needs a small circulation pump.

Electric or gas boost

Many solar HW systems are provided with gas or electric boost. During wet weather solar may not provide enough heat. The extra heating can be provided by an electric or gas boost.

Electric boost - Electric heating elements  will turn on when the water is not hot enough. The heating element can be in the bottom, the middle or both. The advantage of the middle element is that it only heats up half the water.

The problem with electric boost on a solar hot water is that if you shower at night, then the electricity will turn on and heat the water. Next morning there will be nothing left for the sun to heat during the day. This is a tremendous waste of opportunity. You've bought the solar HW system, and you are still buying electricity. The boost should be left off and turned on only if a couple of wet days are expected. The heating can be turned off at the fuse box. Better if possible to have a switch in the kitchen, or somewhere convenient. This will save most of the electricity and change it's position on the chart above.

These systems need a redesign to prevent this waste. A larger  tank holding a few days hot water is the simplest answer. If it is raining for a week you probably don't need as many showers anyway.

Gas boost, the gas heats the water instantaneously just before it is used. Thus it does not waste energy in heating the water unnecessarily. That is why it has a lower CO2 emission than electric boost.

To reduce the need for any boost, see: 

www.energy-without-carbon.org/ReducingElectricBoost

Heat pump

If you have no sunlight then you may need to use a heat pump. This is a refrigeration unit running in reverse. Water is heated by the hot end where the gas is compressed.  It has the advantage of putting out more heat than you put in. The ratio is called the COP, or Coefficient of performance.

COP is more complicated than first appears. For more details see:

http://www.energy-without-carbon.org/EfficiencyOfHeatPumps

Evacuated tubes are more efficient than a flat plate

 A heat pump is an air conditioner run in reverse - reverse cycle. 

Websites

http://www.climatechange.gov.au/what-you-need-to-know/appliances-and-equipment/hot-water-systems/solar.aspx  

Choosing your solar water heater / heat pump - Clean energy regulator

Comparing solar hot water systems - Clean energy regulator

Choosing a hot water system - Your Home.gov.au

Payback time