Hi Andrew,
Welcome to the mostly free world of android!
Nobody is going to bill you for anything without first asking for credit card details or linking you to paypal. Registration gives them an email link so they can send password reminder etc.
Nearly everything on android is free, you can get lots of really good free apps, sometimes there is an alternative premium version for which you pay, this is clearly indicated when you visit the store, usually a dollar or two. Some apps accept voluntary donations, a dollar is about right, though I might donate up to ten bucks if it is a really useful app.
If anything doesn't behave just search for an app or a widget. For example using photobucket directly can be a pain but the app is a real dream. There are lots of utility apps like file managers, advert blockers, book readers, hotmail, photo viewers, and data counter. It's worth trying several to find which works best for you.
Enjoy!
Mike