Hello melseq,
I'm sorry to hear you are having issues with your new iPhone 6. If you are having issues deleting individual messages or conversations, you may want to double-check that you are following the process outlined in the following article:
- To delete a message, tap and hold the message bubble, then tap More.
- Tap Delete All to delete all the messages. Or tap to select the messages you want to delete, then tap
.
After you delete a message, you can recover it only by restoring from a backup.
Open the list of conversations. Then swipe from right to left on the conversation you want to delete.

You can also tap Edit and then tap
:

After you delete a conversation, you can recover it only by restoring from a backup.
Send messages with your iPhone, iPad, or iPod touch - Apple Support
Sincerely,
- Brenden