Is it possible to get the name of your personal blog trademarked?

Is it possible to get the name of your personal blog trademarked. When applying for the trademark, what type of service would it be, how would one go about convincing the system to approve the trademark?

Not likely.

Why? Trademarks are used to legaly register and protected specific product brands. More specifically, products that are involved in local and/or international trade. This is done to protect somebody from making a knock-off product that looks exactly like your product and also has the same name, brand that is. That's why it's not possible to get a trademark for your own blog.