Adding plugin options to WordPress hosted site?

I started out with a free blog on WordPress and recently bought the domain name through the link WordPress offered. I am now trying to figure out how to add plugins.

I assumed that since I had bought the domain name, that I would be able to use plugins and have FTP access, etc. However, after reading around it doesn’t seem like this is possible, and that since I went from a free WordPress.com blog to buying the domain I cannot use plugins. Is this true? I’m very much a beginner at this, but it doesn’t seem likely to me that buying the site through WordPress and having it hosted there has meant less options for me, rather than more. I’d like to know how to set up plugins now that I have registered and paid for the domain name, rather than just a free blog.