Q: How do you use blogger, but have it hosted somewhere else? Do you download the Blogger platform somewhere? Then upload it to another server?
A: All my blogs are hosted by Blogger, so I've not gone through the process. However, it seems pretty straightforward.
In your Dashboard go to the Settings Tab, Then Publishing. Click on Custom Domain. They give you the option to purchase a domain name if you don't already have one, or click on advanced settings if you already have the domain. They will even re-direct from the blogger address you had so you don't lose any visitors.
Q: I'd like to move my blog to a paid web hosting site so this is cool! But then what happens to your traffic, page rank, and your RSS feeds? I'm so nervous about all this!
A: I've not moved a blogspot blog to a custom domain myself, so I'm making educated guesses here but:
Blogger provides a re-direct so any traffic going directly to the blogspot address should automatically be sent onto your new site.
If you are using Blogger's RSS feed, I'm not sure if you'd have to set up a new one or not, but if you are still posting via blogger's back office, I wouldn't think it would hurt your RSS Feed. If you are using Feedburner or some other system, you might need to update some settings.
Page rank would be a whole new ball game. Your existing page rank is based upon your current site with it's set up, traffic, history, etc. The new site given that it now contains all the same information as before should eventually get the same page rank it has now but it would take time. The biggest thing you'd probably have going against it are the backlinks which would still all be pointing to the old domain name and most of those you would not be able to change.
The general rule of thumb is to pick a place and stay there. If you have a well established and busy blog on blogspot, it is probably a good idea to keep it where it is. If you're setting up something new, make a decision at the beginning and stay with it. If you have an existing blog 6 months or more and it is REALLY floundering, you could move it and not be hurt much because you don't have much to hurt. However don't expect the move to solve all your problems.