I can start with either a blog site or register for a static site (with a "blog" type page on that site) The static site being just a bit more complex in setting up?
No..hmm..I'm wondering how I can make this clear to you.
If you want to "register" for a site, you can go to wordpress.com and get a blog account. You're limited on what you can do with it, and it's meant for blogging. However, you mentioned that work uses WordPress, so you're familiar with how to use it to post and such. So I'm going to assume - for the rest of my attempted explanation - that you want to have your own host, with your own domain name, and you're going to install WordPress on your own server and customize it however you need to.
If the above is wrong, then ignore the rest; but if it's the right assumption...
A "static" site is your basic, run-of-the-mill website. You code out your pages, write your content and do all of that in whatever program you want to create your site. It's called "static" because it doesn't change, unless you physically make changes within the file itself and then re-upload and rewrite *everything* on the page - from Doctype to closing body tag.
"Dynamic" is when certain parts are the same, but other parts change as it obtains information form a database. Dynamic sites are great to use because 1) you can do something as simple as a blog - which is the most common and understood part of "dynamic" sites....you write a post, click save, and the site is instantly updated. However you can also take that same technology and 2) use it to manage your entire site seamlessly.
For example, say you have a site that has no interest in blogging. The client has no clue what "HTML" is, much less what to do with it. With a static page, and allowing the client access to update their own site's content, you *always* run the risk of them deleting something important that will destroy the sites' layout - or even worse, functionality. Also, if you have a 100 page site, with the same navigation on each page, you hope to hell no one wants to add a link, or you're spending your Saturday changing 100 pages of code.
Using something like WordPress allows you to create static Pages ("Write Page" in the admin area) to your site. You can use WordPress to handle the layout of the entire site (by taking your design and splitting it up and "wordpressing" it into a theme). Categories and Pages can be automatically generated and listed on the site as you see fit. If you change or add a link, it'll change *sitewide* with a click of a button. You can go in and edit the content of your static Pages with no worries at all that someone will delete or mess with something they shouldn't.
So when you say "register" for either a blog or a static site - no. You get both, already built in, already done. It all just depends on how you code it.
If you want to *start* with a blog, that's why I suggest you place WordPress its own directory, one level below root. Later on, if you decide to built a static site as well, you can still use WordPress to build *around* the blog. You will not have to do anything more than move one file. (And of course, depending on the look of your site, you'll have to code it out as you like.) But blog OR static; or bloag AND static - you have it either way, already built in. It's all just a matter of how you lay it out.
I hope that made some sense to you...