I'm embarrassed to admit this, but this is what resolved my problem. Maybe it'll help someone else:
I would enter my Wifi password in the settings area of the S6. The password window would disappear, and if I tried to open the network again it would not allow me to re-enter a password, so I assumed it was correct and connected (or trying to connect). Yet when I opened Chrome, it clearly said I wasn't connected to Wifi (nor was the wifi symbol showing at the top of the phone).
The problem: it was asking for my Wifi password, which I thought was the same password I could change by logging in to my router settings (at "192.17.143.xy...."). In fact, I could've sworn that's the password I had used to connect my nephew's tablet and other devices to my wifi. However, the S6, although it asks for "Wifi password", it wants your 10 digit alphanumeric code (in my case, my WEP-64 bit key). It looked something like "1XY50B794L" (completely made that one up though).
Sadly, I even consider myself pretty tech savvy. So if you've restarted your phone, router, changed the encryption, and have done everything else, you may just want to make sure that you're putting in the static code and not any kind of personal password. My 10 digit code was found both on a sticker on my router and on the basic security settings screen of my router's webpage. Hope that helps!