It's a good idea to have your blog title on every page of the blog, but it doesn't need to be in h1, h2, etc. It can simply be in your header area. If you have an image use the site title as the alt text. Another method is to set the image as a background and use
<a href="http://example.com">Example site</a> with a negative text-indent.
<h1> tag should be the title of the page, not the site. For individual blog posts, this will be the post title. For your blog's home page, it could be the blog title or something like "Recent posts". Current concensus is to stick to one h1 per page, although HTML5 allows one per
<title> tag should be pretty similar to
<h1> i.e. the post title. It's usually favoured to add the site title to the end, e.g. "My awesome blog post | Blog Name". These two tags (title and h1) are among the most important for ranking.
<h3> etc, you can use as many as you'd like on a page. They are for specifying a document hierarchy. On a blog you would normally use h2 for each major section of your article.