Hello there, DrSmiggles.
It seems that you are trying to send group messages using your new iPhone 5s. The following Knowledge Base article explains how to do just that:
Send a group message with your iPhone, iPad, or iPod touch - Apple Support
To send messages to a group, start a new conversation. Then enter multiple recipients or tap to select from your contacts.
You can add a person to a conversation by tapping Details, and then tapping Add Contact.
You can remove a person from an iMessage conversation with at least four people. Tap Details, then swipe the name of the person you want to remove from right to left. Then tap Delete.
If the group message is using iMessage and has more than three participants, you can leave the conversation. Tap Details. Then tap Leave this Conversation.
Leave this Conversation might not be available if there are only three people in the group iMessage or if the group message uses MMS.
You can name group iMessage threads. While viewing a conversation, tap Details, in the upper right. Then drag down and enter the name in the Group Name field.
If you don't want to get notifications for a group message, tap Details. Then turn on Do Not Disturb. You'll still receive the messages in the Messages app, and you'll still receive notifications on the lock screen.
If you send a message to multiple people and at least one person isn't using group messaging, an individual message will go to each recipient using either SMS or MMS. Only the original sender will get any responses to these messages, in separate message threads.
To use group messaging with MMS, you need a plan from your carrier that allows sending MMS. Carrier messaging rates may apply.
Check with your wireless carrier to see how many people you can include in a group message with MMS.
You can turn off MMS group messaging in Settings > Messages > Group Messaging. You can still send and receive group iMessages.
Thanks for reaching out to Apple Support Communities.
Cheers.