I am the web master for a local professional organization. They already had their stuff setup w/ GoDaddy (and didn't want to change) and I converted them to WordPress about two years ago. Prior to that, the had some sort of Mac WYSIWYG thing. Personally, I find the GoDaddy website very unintuitive to navigate through, but I seldom have to use it. The WordPress install functions like an independent install in terms of themes, as far as I've experienced.
One weird thing I've noticed is that I run into trouble if I update WP through the dashboard. Instead, I have to log into GoDaddy an update it through my web apps. Not a big deal, but the first time I upgraded through the WP dashboard my site got really slow until I also updated it through the GoDaddy menu.
For my personal site, I use ICDSoft. I've been using them for a decade or so, and I really like it. It's not the cheapest ($72 per year), but you get 1 GB storage and 20 GB bandwidth. I've been grandfathered in at a lower price, too (I think I pay $54). To be honest, I just don't want to bother moving it. If I were starting today, I'd try A Small Orange. Oh, and I have my domain names at namecheap.