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Will my contract automatically renew or will I get some form of message that will ask me whether or not I wish to renew my contract? Even thought it will only be finished in the next few months, I am planning on getting a new device within the next few weeks (my current one is already paid off). Will my new device continue to be paid off when the new/renewed contract begins?
The contract will simply expire. Unless you call to make changes to the plan, or purchase a device through Telus, the existing plan will remain unchanged. Many people these days opt to buy their own phones to keep the old plans instead of getting the more expensive ones that are required with the new devices because of the shorter 2 year terms in which the phone must be paid off in.
Once your contract ends you are automatically switched to a month-to-month plan. You can continue this indefinitely,as @Nighthawk says, or, if you want a new subsidized phone, you can enter into a new contract, on what ever terms Telus offers / you negotiate. Telus will not subsidize a phone on your current plan, as there is no defined end point, thus they can't recover the subsidy they offer.