So what happens after one year if I want to keep my domain but change
You're free to either point the domain to another hosting company, or even transfer the domain to another Registrar, providing you don't let the domain expire, or enter the redemption stage after it expires.
Does my domain name belong to Godaddy now?
No. When you're listed as the Registrant for a domain (i.e., your contact information appears in the WHOIS record for the domain) with an ICANN-Accredited Registrar like GoDaddy, you have all the Registrants' Benefits and Responsibilities that come along with that as provided under the Registrar Accreditation Agreement (RAA). See this question for more about that.
Or can I just go and buy it again from another hosting company or even
get it for free if it comes with a package?
After you register the domain with GoDaddy as part of their promotional hosting package, you'll need to transfer the domain if you're trying to take advantage of another promotional package elsewhere. Transfers to other Registrars are often at a discounted rate to earn your business, but you'd have to find a promotional package that includes a domain transfer versus a new registration.