You can wipe the iPhone and start over:
Restore your iPhone, iPad, or iPod to factory settings - Apple Support
Or
You can sign out of your Apple ID from all services:
Settings->iTunes & App Stores->Apple ID
Settings->Account(top of settings)->Scroll Down to "Sign Out" for iCloud.
Settings->Messages->Send & Receive->Apple ID
Settings->Facetime->Apple ID
Then you can go to any of those locations and Select Create new Apple ID in the Sign in Menu.
Sign in with your Apple ID - Apple Support
Note that doing with will not delete any content, and any Apps downloaded with your other Apple ID will remain tied to it, and require it to update.