Use Hostgator, or another web hosting service.

The domain name mapping feature works, but won't give you control over the plugins and things you use. You can change the themes, but it certainly won't provide the freedom a self hosted site will. I haven't used it in a while, but as far as I know you are limited to a small number of themes. Only some of those themes will allow for a custom header, which will result in your site looking like millions of others out there.

I'm a WordPress designer, and a content writer, so let me know if you need anything.

I would prefer bluehost for hosting wordpress sites

If you're planning on monetizing your site at all, you need to be self-hosted. WordPress.com doesn't allow commercial activity and they will trash-can your site when they find it.

What are your plans for this site? Monetize with a custom theme? Just for fun/as a hobby?

I am currently using site5.com to host some of my clients blogs and all works OK.

Short answer, yes it is better to get your own hosting. Also make sure you register your domain with a provider that is not your web host! Never host your website and domain in the same place.

It is possible to change a theme, I have tweaked themes myself. If you are looking for a completely custom theme you would be better off designing and building your own from scratch or hiring a qualified WP Theme designer to do it for you. If you want to learn how to build your own WP Theme, check out WPdesigner[dot]com