Backup iPhone contacts etc to move to new iPhone (no iCloud)
Switching Apple IDs

If you go to Settings > iCloud and Sign Out, everything should stay safely on your phone with the exception of iCloud Drive documents and photos from your Photo Library. If you were to then sign into a different Apple ID, such as a newly created one, you could turn things like Contacts and Calendars. I say should because if you get any warnings saying something will be deleted, you should take them seriously.

Unfortunately , this requires taking extra precautions for iCloud Photo Library and iCloud Drive. If you do have things in iCloud Drive or iCloud Photo Library, you can go to www.icloud.com on a computer and perhaps download your documents and photos from there so that they are safe on a computer.

The Alternative

Contacts and Calendars are a tricky thing to backup without iCloud as they don't seem to be included in the catch-all iCloud Backup.
There may be many alternatives to the above, but the only one I thought of off the top of my head was to check with your carrier about using your SIM card for that purpose.