My Google results are really great right now. I am concerned that it will affect those results.
The authority and page rank your blog has earned on the .wordpress.com sub-domian belong to that sub-domain and cannot be transferred. When you purchase domain mapping the domain URL starts from zero. All the content will be re-indexed by the search engines and in about 4 months time, all things being equal, the domain will have the same authority and page rank the sub-domain had.
In your case http://mrsarsenault.wordpress.com/ is unranked and therefore there is no pagerank that will be lost.
Also, will everyone who has linked to my blog suddenly not be linked to my new blog?
You will have to notify those who have linked to the sub-domain URL to change the link to the domain URL.
Will it forward the old blog automatically to the new one?
If your blog remains free hosted here at WordPress.com then there is no content moving involved at all. What domain mapping does is seamlessly transfer all visitors who click links to your blog content by using the original sub-domain links to the new domain links.
Note that all you are purchasing is a domain name and domain mapping. Nothing else changes: http://en.support.wordpress.com/domain-mapping/register-domain/#before-upgrading-important-notes