Hey Nicky,
Please note that just because youve deactivated the phone from one carrier, the data is still on the phone and can be backed up (Through iTunes or iCloud) At anytime. When you sign up for the new carrier, they will gather information from your current carrier, which usually is account number, billing address, last four of social for primary acct holder and obviously the phone number you wish to port. Once the port goes through, you should immediately have service on your new carrier with your existing phone number. (Note that sometimes ports can take up to 10Hours, however most of the time they happen instantly)
When the port goes through, it automatically cancels your old carrier plan which results in your having to pay a ETF (Early Termination Fee) if you were still under contract. i.e. you sign a 2 year contract with AT&T, the 2years havent passed and you decide to switch to T-Mobile, when you switch to T-Mobile, they will port your number and that automatically cancels At&t which results in you paying a ETF, if you werent under contact no ETF.
As far as carriers with no contract, it would depend on where you live and the Coverage. T-Mobile is starting this initiave of the "un-carrier" by having no contracts, BRing your own phone kind of deal, they will even pay your ETF if you switch to them, So look into that.