While it's true that some bloggers host their blog on their own server, you don't have to. Blogging is remarkably convenient—you can do it from your cell phone—you can sign up for an account and start blogging in minutes. (This blog has showed you how to do that with Tumblr.)
This is a list (but not an endorsement) of platforms that will let you get started:
The involvement required from you varies from one service to another. Make sure you explore your options before committing to anything!