There are a few different routes you can use to start a blog, depending on what you would like to do with the blog. If you are looking for a free service online that you can just sign up for a go,
Blogger is a good spot to try. You sign up for your blog, enter a few setup options, and you are ready to start posting to your blog! You will have a username and password to log in with, and you can then send your family, friends, and the rest of the internet to your blog to read your posts! Blogger will let you customize the look and feel of your blog as well as certain items contained in the margins. If you are looking for a more advanced blogging experience (bordering on full site management), WordPress is a very popular piece of blog software. You will need a webhost with php and mysql capability in order to install WordPress (See
How do you set up a Content Management System?), and once you do you will find many templates and customizations available to you to design your site and accomodate it as it grows.