If you're after free hosted there's really two main choices people go for -
Blogger or
WordPress. I began with Blogger about 2 years ago and I'm still using it - I like its simplicity (the newest version is a piece of cake). If you're brand new to blogging there's no harm in starting with Blogger to get the feel for it. Another advantage of Blogger is that you can easily integrate AdSense to try and earn a little advertising revenue.
Free hosted WordPress blogs are the other choice people usually go for - they aren't quite as simple as Blogger and you can't use AdSense with them (might be an issue) but they do look more professional and have greater functionality.
Whichever route you choose it's advisable to use your own hosting and domain name once you've got the feel for it. The most popular paidhosting blogs use WordPress. Using your own domain also gives you the advantage of being able to use AdSense and you can easily customise the appearance of your blog by downloading free themes (like skins). It's also quite easy to move your hosting from one provider to the next.
Hope this helps!