The quickest and easiest way to get your blog up and running is to use a hosted solution. What does that mean? Essentially using a website to create a website. It means you don't have to be technical and there is minimal setup to get started.
Blogger: Blogger gives you the option to create your blog for free and host it with BlogSpot. The downside to that, the required navbar that will display at the top allowing visitors to click the the next and previous blog hosted by BlogSpot. First off, that just looks unprofessional - re: cheap. Second, you don't have any control over what blogs will show up when the user clicks those link. Yeah, think about that for a minute. There are hacks for how to "hide" the navbar but it probably is against the terms of service. Your other option with Blogger is to use it to generate your blog, but have it FTP to a host that you pay for. That will require a little more setup and technical knowledge on your part. In addition, according to the terms of service, you must include a link back to blogger as "payment" for using the system.
Typepad: No free option with Typepad but the basic package starts at $4.95. There is no advertising via Typepad so you have complete control over the look and feel of your site. No worries about links from your blog going someplace inappropriate - unless you put that link in! No requirement for linking back to Typepad either.
Blogger: You create your blog using a form to name it and create the URL if you are hosting with BlogSpot. Then you can choose from a handful of templates which you can later go back and customize it. There seem to be more choices for templates after you've done your initial setup. And you can go in and create your custom template too if you have enough technical ability.
Typepad: You can create your blog using a form to name it and create the URL much like with Blogger. There are lots of options for your website look using their templates. Lots of different layouts and and color schemes. And you can customize your template colors and header (like I have for this site).
Seriously, they are pretty close in features. One big difference is that Typepad supports trackbacks and Blogger does not. Why do you care? Trackbacks are one of the ways you can let others know about your blog by sending a ping to the blog you reference letting them know you referenced them. So then on their blog they may show a list of the blog posts that refer to that post, and you've got a link to you! Learn more about Trackbacks in TypePad's Knowledge Base. So, you don't have that type of notification using Blogger which means you miss on a big opportunity to connect to other sites.
The other big features missing from Blogger is the ability to add categories to your blog. You can type in labels but the nice thing about Typepad is you can setup your categories in advance and just pick them from the list when you create your blog post.
On Blogger's side, they do have built in search via the NavBar (of course, they are owned by Google so you'd expect this). However, there are lots of ways you can add search to your blog using many different third party additions such as a google search or technorati search.
So, check them out and see what you think. My personal preference is for Typepad. I feel that you can get a professional looking blog with minimal technical knowledge for a very reasonable price. Typepad blogs are easy to setup and there are lots of widgets you can use to add functionality to your blog. And best of all, no advertising. If you'd like to read a MUCH more thorough comparison between Blogger and Typepad check out the How To Blog post.
And yes, I know, there are ways to hide the NavBar in the Blogger BlogSpot hosted sites, but it probably goes against their terms of service so I don't recommend it.