If you’re serious about blogging, I strongly advise you to avoid using Blogger, WordPress.com, or *shudder*, LiveJournal. The first thing you should do when you start a blog is get a domain name. At $7-$10 per year, it’s not going to empty your bank account, and it will be a good investment. (Not to mention that people will take you much more seriously if you have your own domain) You could point your domain pretty much anywhere. You could use it with your Blog*Spot or WordPress.com blog, but I would advise against it. While Blogger and WordPress.com are most likely the two best “free blog services” available, they’re limiting in terms of what you can do.
If you pay for a cheap shared hosting account (1and1 has a $3.99/mo plan, which is good for a beginning blog), and install a copy of the free WordPress blogging software (that’s WordPress.ORG, not .com!), you gain much.
By hosting your blog yourself, instead of relying on a free service, you gain the ability to customize your design in ways impossible with the free services, to use WordPress plugins, run your own ads, and you’re not dependent on the service. If WordPress.com decided to charge a subscription fee instead of providing the free service they’ve provided, you’d have to pay or abandon your blog, pretty much. What if they changed something about their service, and you didn’t like it? (An example would be adding a big, forced banner ad to the top of your blog.)
If you’re using a free blog service, you can still make the switch to a self-hosted blog. WordPress has the ability to import posts from Blogger, WordPress.com, LiveJournal, and a few others (including RSS). However, you can lose some data in the transition. Depending on which service you use, you may or may not be able to keep the comments on your posts, for example.
Free blog services are great for personal blogs, and experimenting with blogging before deciding to commit to it, but if you want to run a serious blog, I highly recommend going with a self-hosted WordPress installation. At the very least, if you have a tight budget, get a domain and point it to your free blog. That way, if you decide to go the WordPress route, you will be able to move and keep your readers and backlinks.