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.
It can be very difficult to change hosting services. It is sad that hosting cannot help to transfer website files from one hosting to another. I tried to do it myself, but as a result, I broke something and now my site does not work. Do you know if it is possible to ask a web hosting service to help transfer all important files? Because I don't really want to figure out what my problem is. My friends said that the problem is most likely in DNS and I was even able to find a kind of very useful article about DNS https://www.softwaretesttips.com/dns-probe-finished-no-internet/ , but I very much do not want to look for the problem in DNS .