I've had 4 moves, all have gone well. For peace of mind, take a photo of the final reading. The process is simple - tell your new supplier you want them to supply, set up direct debit, give them a meter reading when they ask for it, they will tell British Gas who will send you a "sorry you're leaving" letter. When your new supplier passes the final reading to British Gas, British Gas will send you separate "final bills" for both gas and electricity, and either take a final direct debit or send you a cheque for what they owe you.
Be prepared for the saving not to be quite as large as the comparison site tells you. If you've never switched, it's worth switching, but after that you probably don't need to do it every year. Remember too, that if you're on a fixed term tariff, the comparison assumes you'll drop on to an expensive "standard variable tariff" at the end of the period, so the saving is against that scenario, not against continuing at your current rate for the rest of the year.
If you move to a fixed price tariff, time your switch to the next one carefully. It takes about 2 weeks to switch. Move too soon, and you lose the benefit of the last week or two of your old tariff, leave it too late and you'll spend time on an expensive variable tariff.
They review your direct debit after 6 months, so if you move at the start of winter/high usage, they'll up your dd just as you're moving into summer, and you'll build up a big credit. So much for dds evening out your bills across the year.
Even British Gas was reasonably efficient with the change over, although their final bills are confusing, to say the least.