How can Google Analytics (or any other analytic tool) be leveraged to improve SEO for my blogging site?

Google Analytics will help you analyze the best time to make a post. This in turn will help you reach the maximum audience which will in turn help make Google believe that your blog is worth indexing. Apart from that the kind of audience (geographically) you get will help you re think or improve your writing or the topics you choose. It will also tell you the links that are most useful to you. In the sense the backlinks that are actually generating traffic to your blog. Also keep a check on the bounce rate. I am sure you want your readers to stick around. There are many tips on how you can do that. Eg: include links to other topics within the blog in a post. I hope this gives you an idea on where to start. Keep exploring Google and Google analytics. :)

Most important metrics that any first time blogger should focus on are traffic source/medium & their bounce rate. Generally blogging sites have high bounce rate. You can use Event tracking method to see how many visitors are reading you entire blog post. If you are doing any campaign on social media, use URL builder tool to track your visitors. Hope this helps.