Electricity and gas - just tell the current provider the date you moved in and give them the meter readings. Rather than tie yourself into any contract you can just request that you go onto their standard rate and then look into things properly. With some providers like EDF you can go onto a lower tariff with no tie in period or cancellation penalty so you may as well do that from the off. Once you have a handle on your consumption you can shop around.
Typically, unless you have an electric boiler or big old immersion heater and an electric cooker when you previously had gas, your electricity use is unlikely to change much. If you move to a bigger house with gas central heating then you gas use is likely to go up, but it also depends on the age of the place, the insulation, etc.