If a friend wants me to build a web site for his plumbing business, how do I set up a hosting package? Do I purchase an account from a web server or do I make him do that? If I purchase the account, do I give him access? What do other people do?

In my experience, I have suggested purchases, keeping the client informed (sometimes bugging the client at every step along the way, and maybe too much for the client's patience, but enough to be ethical on my part). I had the client physically present for the web hosting purchase (to enter e.g. credit card info including validation number on back of card) and to enter an agreed-upon access password, for both of us. We printed website access info (logon info, FTP access info, etc. Most assuredly, the client had access, in case my services prove unsatisfactory, or in case the client proves unsatisfactory to me and (for either of these reasons) we decide to quit doing business with each other...

use a hosting service if they charge a fee of 11.99 a month bill him 15.99 a month and charge to set it up!

It's best to pick out a good web hosting service that meets his needs and then sit with him and help him sign up. Everything should be in his name, including the domain. Pretty much the only thing you should share with him is the password to upload files and maintain the website.