You will need to add a data plan to your Galaxy S3 and only then will you be able to tether by USB wire. I wouldn’t bother with Bluetooth tethering. Their data plans start at $30/month for 2GB.
Pretty much all android phones sold in the last 5 years from Verizon will allow you to use the phone as a wifi hotspot. Verizon wants to charge you an extra $20 a month to do this. There are apps that allow you to share your wifi without paying $20 for data you are already paying for. In my experience these apps need root access to your phone. I have been doing this with my Verizon smart phones for years but not the Galaxy S3.
Since you have Verizon, note that Verizon can not charge you for tethering:
They can if you subscribe, but if you tether on your own(through a app - usually requires jailbreaking or rooting which also are both legal), they can’t do anything about it.
Seems like the spectrum they squired from the TV networks comes with strings attached.
The app I use for tethering (Easy Tether) works without having to root or jailbreak the phone. I’m not sure if there’s a WiFi hotspot app that does the same.
On the other hand, if you have an unlimited data plan, apparently Verizon thinks the FCC ruling doesn’t apply to you and wants you to pay a $20 fee to tether:
I don’t know why that is the case. Their statement wasn’t clear at all.
(And in practice, you can easily tether and fly under the radar as long as your usage doesn’t go up too much. And with a capped data plan, it doesn’t matter anyway.)