Like most things you do in Weebly, creating a blog is quite simple. When creating a new page from the "Pages" tab, simply select "New Blog." Make sure to add the page before naming it. If you name it first, you will be renaming the current page.  I found that you can set the blog page as your home page by dragging it to the top of the page list.  Blog posts are similar to standard pages with a few differences. .

Weebly conveniently saves everthing as you go. Nothing in your blog editor gets published until you click on "Save to Live." No need to click on "Publish" after this, the new text is already posted to your site. To wrap text around a picture in the blog or to add video, drag one of the content blocks from the top to the blog entry while in edit mode. You can then delete the default blog text block. Also note that you can include a "read more break" by dragging this option from the top. The "Manage Blog" button opens a window for managing blog comments and setting your default settings for comments. You can change these settings on a post by post basis by choosing "settings" for a post. From here you can also change the name of the of the blog post's html file as well as send a trackback. A trackback is way of notifying a blogger that you referenced their blog post. It also lists your reference in the orignal post's comments. For more details check the Ultimate Trackback Help Guide.
You can create categories for your bog posts by adding categories below your blog post. Seperate with a comma to add a post to more than one category. The categories you create will show up on the side of your blog.