Wordpress is great for blogging software. You can get a free account at wordpress.com and try it out if you like. The downside to that is your domain name would be
myblogname.wordpress.com.
If you want a more professional-looking URL (e.g.
www.myblogname.com), you can register a domain and get a hosting solution with Wordpress blogging software pre-installed from a company like Lunarpages or Dreamhost. I've used both in the past, and setting up Wordpress on either is super-simple. IIRC, hosting plans with unlimited bandwidth go for like $5/month or something stupid cheap these days.