You may not actually need to switch internet providers just because you're moving. Most ISPs let you transfer your service from one address to another. Always let your ISP know you're moving regardless—if you move to an area where your current ISP doesn't offer service, it may waive your cancellation fee.
When should I cancel internet when switching?
Now that you've moved on, you need to cancel your old internet service as soon as possible. Make sure you don't call customer service before you've already found a new service provider. You don't want to be between internet service providers, so try to schedule your cancellation a little after your new service begins.
What happens when you change internet provider?
You won't normally lose service. Switching broadband suppliers is a lot smoother than it used to be and, in many cases, you should be able to switch with little or no service interruption. Your new provider will advise you when the switch will happen and if you're likely to experience any loss of service.