How do I update a 3rd generation iPod to iOS 7

You don't. A 3rd generation iPod Touch cannot be updated past iOS 5.1.1.

If you have an iPod touch with iOS 6.1.6 you have a Fourth Generation iPod Touch, and it cannot be updated past iOS 6.1.6

Only a Fifth Generation iPod Touch could be updated to iOS 7 when it was released. Now it can only be updated to iOS 8.

That's really unfortunate. Do you know if there is any way to install an app that requires iOS 7 to a 3rd gen iPod? Is there a way to install the last compatible version of an app instead? I know that is an option when you go to update an app that now requires iOS 7; the app store says it can't update to the most recent update, but it asks if you want to update to the last compatible version of the application. Like I said though, that is only if the app is already installed. Nearly every app I try to install now fails because I don't have iOS 7. Is there any way around that? What if I jailbreak my device? -Ray

If the developer supports it then download the current version of the app on your computer's iTunes so that it's in your account's purchase history, then on your iPod go to the Purchased tab and see if you can download an older compatible version of the app - you should see something similar to below if you can :

If the app is paid-for then check with the developer first to see if that will work, don't buy it and then find that it doesn't.

Jailbreaking can't be discussed on here : Unauthorized modification of iOS can cause security vulnerabilities, instability, shortened battery life, and other issu…