IPhone 5 not sending text messages, not able to pull up Safari, acting strange?

I can't send texts. It just says "sending…" forever. Once in a while, one will go through, but I usually have to cut off my phone, cut it back on, and resend it. A few minutes later, it resumes not sending texts. Have to restart the process. It says "searching" where the carrier name appears randomly. My service is fine. Other people around me have my carrier (Verizon) and have no issues.

Whenever I try to pull up Safari, going to Google for example, it pulls it up, and after I type the first letter it freezes. Then it flashes the keyboard over and over again for a few minutes, and eventually shuts itself off.

Every app I have is running very, very slow and acting glitchy. I only have a few apps:
YouTube
Twitter
Tumblr
Instagram
The Weather Channel
And I've had all of these since the day I got my iPhone. They've never caused any issues. As a matter of fact, the only one I really use is Twitter. I rarely open the others.

What's wrong with my iPhone 5 and how do I fix it?

I think a software update can help.

Try to update your software to the newest iOS 7.1.2 and hold the screen shot buttons until the phone shuts off.