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It seems like a new version of Babel was released in August 2018 and it is what all the latest versions of our tools are going to need, so we might as well upgrade it now.
What actually spurred this change was looking at lingui for #12; it requires that we use Babel to convert JSX syntax into JS, rather than relying on TypeScript to do that for us, and Babel 6's JSX transformer doesn't support the new React fragment syntax that we've been using. So this PR will allow me to experiment with integrating lingui into the project, but it's a good thing to do regardless.
To do