The word blog is used to describe different things in different cases. If you want to build a blog like website in Joomla there is no problem to do so.
Joomla is extremely extensible and there is virtually no limitation what type of website you will build with it. A Joomla Blog can be extended using all other functionality Joomla provides. With a Joomla based blog you can have a forum, shopping cart or a multi functional gallery all bundled in one solution. Checkout these helpful tutorials for more information how to build your Joomla website with Joomla 1.5 and how to setup a Joomla 2.5 web site.
Still if you would like to build a blog website there are much easier solutions that are designed specifically for that purpose. The SiteGround recommendation is to use WordPress in such cases. Various blog tutorials can be found in the How to Create a Blog Site section.
WordPress is a powerful software that is easy to work with once you have the right host to support you. Before you lose any more time with the wrong host, check out the SiteGround WordPress hosting offer.