Choosing the domain.  You will pay about $10 for the year. That's it.  Don't spend any more!
I use http://mydomain.com but godaddy, enom, or any of those will do.  You can wait and get your domain via Google when you sign up for Google Apps this makes some things easier like adding a Cname listing for new sites but you also want to have other domain management tools you can use. See if you can look at the domain management of the provider to see if you like it.
In choosing the domain hosting, pick one that has domain management tools plus url forwarding. You don't need email from the domain host if you are going to use Google.  You just need the ability to set your mail servers.  Keep in mind that the domain is just a listing.  It's like getting your name into the phone book. It doesn't mean you have any web space just like the listing in the phone book doesn't give you a phone.  That's fine though. Don't be fooled into thinking it's only a "real domain" if it's packaged with web hosting.  Your domain is no less real than a phone listing that you didn't pay the phone company for.

Setting up your domain: Here's a sample of how to set up your domain management. Basically you are setting a CNAME to the domain name server ghs.google.com. for any domain or sub-domain that Google is handling.
URL forwarding: You may want to set some sites up with URL forwarding instead of CName.  For example I set http://admin.ofYork.com with URL forward to take you to http://www.google.com/a/cpanel/ofYork.com since it's not really part of the public site so it doesn't have to appear that way.  The difference between the two as far as it appears to the public is that CNames are like a DNS entry for the internet and point to the site leaving your address in the address bar of the browser while URL forward will either change it to the new address or mask the real address by forcing the browser to still show the address you set but it may stay that way even when users browse away from the page.

Setting up email in your DNS Management: You can see my set up for Google's mail servers here in the MX records.  Google has a good help on this that you can get to from your Google Dashboard or control panel. http://www.google.com/support/a/bin/answer.py?answer=86166 Or this link.