I want to change the text and the look & feel of the "Register" button to this:https://www.screencast.com/t/MLiXZKX1
How would I go about finding where this HTML is to make the change?
You can make a lot of changes in Vanilla with css.
Use a web inspector to see what rules are applied to the element in question.
You can make CSS changes in your theme's custom.css file, or by using the CSSEdit plugin.
You can change the text by using a custom locale file and adding
$Definition['Register'] = 'Any text you want';