I started out with Blogger, but migrated things to wordpress a little over a year ago and never looked back. But I do still have a blogger account, and probably will start a new blog there, just as a personal blog.
If you're blogging just as a creative outlet or as a personal blog that you want to share with family and friends but aren't concerned with trying to be a huge web presence, Blogger is quick and simple. You can also monetize your blog. But Blogger also has fewer customization options, so if the design is extremely important to you, you may not be satisfied there.
There are two options if you want to use Wordpress. Wordpress.com provides free hosting for blogs, and of course uses the Wordpress platform, but there are restrictions, not the least of which is that you can't have ads or sell products from the free sites.
You can also use Wordpress on a self-hosted blog, which will give you the most freedom in terms of content, monetizing, and so on. There's a little more of a learning curve, but it's far from impossible to learn, even if you're not a techie. Most webhosts will install the software for you, so all you have to do is navigate the dashboard.
If you're not sure what direction you want to go with the blog, why not start with a free site on Blogger or Wordpress.com and see how it goes? Moving a blog from one platform to another isn't terribly difficult. (I don't know about Blogger, but Wordpress has pretty easy-to-use tools for it.) No matter which site you choose, definitely buy your own domain name, and have it pointed to your blog. If you do that, you can move your blog as often as you want and don't have to worry about losing readers that can't find you. If they're going to JoesBlog.com, they'll never notice a move.