how to get notes from icloud to new iphone.

You're welcome.

If I understand this correctly it will the the only remedy for the following issue:

I have two sets of notes: one on my iphone and one in the cloud. There obviously is no other way to transfer the iCloud stuff permanently onto my iphone other than copy and paste them.

The real problem behind this: if you disconnect the iCloud the respective notes disappear from your iPhone.

To rephrase the question: how can I make sure that I see all of my notes when the iPhone is offline.

Notes and other data synced with iCloud remain on your device even when it's offline. A cached copy of your data remains on your device; any changes to the synced data will be uploaded to iCloud when the device connects to the internet again.

The way you explain it is just what I would expect to happen. Unfortunately the situation on hand does not fit in with this.

I do have notes in the iCloud which disappear from my iPhone as soon as I go offline. Is there any way other than copying and pasting them into new notes to get them on the iPhone permanently?

If that's happening something is wrong. You might want to try deleting your iCloud account, then adding it back. To do this, save any photo stream photos to your camera roll, then go to Settngs>iCloud, tap Delete Account, provide the password when prompted (if running iOS 7) and choose Delete from My iPhone, then after a minute of so sign back in. (This only deletes the account and your iCloud data from your phone, not from iCloud. Your data will reappear on your phone when you sign back in.)

Unfortunately, there is no mechanism for copying of moving notes from iCloud to store them locally on your phone. You have to copy and paste the text into new notes created in the On My iPhone account.

Solved by copy/paste.