The reason one purchases a domain name and domain mapping upgrade is because when any links to the original content are clicked anywhere on the internet they seamlessly direct visitors to the same content under the new URLs.
And if I change my domain name, will I be able to add some more new widgets that I would like to use?
No. All that changes is the URLs.
Important Notes Before Upgrading http://en.support.wordpress.com/domains/#important-notes-before-upgrading
All About Domains http://en.support.wordpress.com/domains/all-about-domains/
You must be logged in as Admin http://en.support.wordpress.com/user-roles/#administrator under the exact same username account that registered the blog to access the blog's dashboard at > Store > My Upgrades.
In order to map to a domain one must have an underlying .wordpress.com subdomain blog to map from and the domain name you desire must be available for purchase, or you must already own the domain URL.
As you have registered a .wordpress.com blog you log in as Admin under the exact same username account that registered the blog and purchase a domain name by purchasing an annually renewable domain mapping upgrade for the blog.
See here if you do not own a domain http://en.support.wordpress.com/domains/
See here if you already own a domain http://en.support.wordpress.com/domains/map-existing-domain/
Dashboard > Store > My Upgrades is where you do the primary blog set up knowing it can take between 24 - 72 hours for domain name propagation to take place throughout the internet. You can view the DNS changes here > http://www.whatsmydns.net/ What's important during that stage is to be patient.