The name of hosting provider actually doesn't matter much since the problem of how to transfer web hosting from one company to another has a common solution. Thransfering FROM GoDaddy is almost the same as transfer website to godaddy which you probably already did in the past.
Anyway, the actions you need to perform is:
1) move your site's files to a new hosting
2) move the database to a new hosting (it can be easily done through web interface which must be in any hosting service nowadays)
3) redirect the domain DNS to a new hosting
Still, the DNS change can take up to 48 hours, so there is no guarrantee that your site won't get offline for some time if you've less than 2 days remaining on GoDaddy.