Hi JessicaMendenhall1994,
It sounds like you are asking about three issues regarding your Messages app on your iPhone. First, you are unable to send Messages while on Wi-Fi. Second, when you send a Message to another iPhone user, it is sent as an SMS text (green) instead of an iMessage (blue). And finally, you are not receiving notifications when you receive a message. Is that right?
Jessica, have you tested to see if the issues persist when using different wireless networks? Try restarting your modem and router if the issue is with your primary wireless network. Test using cellular data instead of Wi-Fi.
The resource below provides some basic troubleshooting steps for the Messages app. I've highlighted a few key items from the article here. Click to see complete details:
If you can‘t send or receive messages on your iPhone, iPad, or iPod touch
- Make sure that you’re connected to the Internet through cellular data or Wi-Fi.
- Go to Settings > Messages. Check that the type of message that you're trying to send is turned on (iMessage, SMS, MMS, or Group messaging):

- Restart your device.
- Check for a carrier settings update.
- Check for an iOS software update.
The resource below will provide some troubleshooting steps to address the issue you are having with Notifications for your Messages app:
Use Notifications on your iPhone, iPad, and iPod touch
Take care,