Whatever remaining time that is on your domain follows the domain. That means if you have 2 years left on a domain with GoDaddy and you move it to somewhere better you SHOULD have the same time remaining on the domain.
Once you transfer the domain, that's where it will be registered, so the next time you need to renew your domain, you will pay your new registrar for the renewal and not GoDaddy.
However, I do want to warn you, transferring domains can be a real hassle. You will be required to obtain a key from one registrar and provide to the other, click links in emails and then some emails may have links that you click only if you want to cancel the switchover. Just make sure you read over everything you receive. It's important to know understand the process of leaving GoDaddy and the process of coming to your new host as all registrars are different.
With some registrars, you can unlock your domain for transfer in the user panel, but one registrar I've used required me to email them a request for the domain to be unlocked.
Best of luck!