Typo3 Blog Extension:
Unfortunatly I have no experience with the typo3 blog system though, it's praised a lot on different pages to be quite easy. Furthermore typo3 is a community driven system, meaning there is to be expected new features, extensions and so on.
How to use the typo3 blog:
On following homepage, there is are links to tutorials how to use the typo3 blog system. You can also find a video tutorial.
http://www.goldenplanet.com/products/typo3/extensions/blog.html
www.blogspot.com
www.blogspot.com is an "external" blog which is quite famous. It is easy to register and to configure. As example you can use simple Drag&Drop to design your own Homepagestyle.
Unfortunatly I don't know the style of your actual homepage. If you want to "include" the blog in your homepage which is set up with typo3 cms it is of course "better" to use the blog extension of typo3. Also if you are going to stay with this content management system. If you don't want to spend too much time, on learning how to handle the typo3 blog extension or want to use the blog as an external link I recommend an external blogger such as www.blogspot.com