Backend Framework - Express is nice and minimal. Sails is built on top of Express, is heavier, follows MVC architecture and gives you a lot of things pre-configured.
Template Engine- I have used ejs and jade, both are good.
Frontend- React, although its just views, because it has a very strong community and resources. There are many other client-side options like Angular, ember etc. Since you want to be a full stack developer, you should consider trying out isomorphic apps(client side and server side rendering). For which, React+Flux is a great combination. This post engineering.devmag.io/post/1/creating-an-isomorphic-blogging-app-using-react-and-flux explains it a lot more detail.
Tooling- I find gulp and webpack to be pretty good.
ES6 Support- No, you don't need to use babel. There are other ways to use ES6 in node like using --harmony flag in node, but it has limited support for ES6. So Babel is definitely the recommended way.
Hope this helps. :)