Upgrade from CS6 to CC?

To add to this: there can't be "upgrades" from a perpetual license to a subscription model. What Adobe does is to offer incentives for new subscribers for the first year. Even if that is an offer for old customers, it's not considered an upgrade, which also means that you keep ownership of your current CS6 license and you may use that independently from your subscription.

Believe me, the value of such a license is quite low. I'm still the proud owner of a CS5 package, upgraded from the very beginning of the Creative Suite and from the single program packages before, and objectively I will never use that license again.

FYI: with a CC subscription you have access to all the old versions up to CS6. This is especially useful, if you need to use a program that is no more part of the current package, because it's EOL (Fireworks, Encore, ...). Support, however, is limited to this user to user fora.

Or do I need to buy Creative Cloud and pay that monthly subscription fee to get the newest versions of Photoshop, Illustrator, etc. even though I already own the CS6 versions?

Yes. There is no connection between your ownership of CS6 and the Creative Cloud plans.

Just select the Cloud plan you want and join up.