Your theme is displaying the "like" button correctly and the button cannot be moved to another location. Unless you use the P2 theme which is a Twitter-like theme, the front pages of our blogs showcase content and comments are collapsed on that page and you cannot change this. The way the comments function is coded into the core of WordPress. By clicking the post title or the comments link the posts on the front page open on their own page where all comments display.
In a new blog one does not have many posts or comments. In an established blog if the comments did not collapse on the front page it would become a mile long in no time flat and page loading time is a page ranking factor. We have only seconds before visitors refuse to wait and click out.
If you are having trouble showing the sharing buttons on your blog, please check the following:
I can’t see the buttons on the front page.
Make sure to select “Posts and index pages” for “Show sharing buttons on” setting. http://en.support.wordpress.com/sharing/#troubleshooting