How many (and what kind of) email accounts should I be looking for?
How many email accounts you want to have is up to you... only you know how many family, friends, or co-workers you may need to set up in the future. As for number, there are generally two types of plans. One email, and "more than you'll ever need".
As for what types of email, there is the type where you use a specific program to read your mail like Outlook or Eudora. These formats are called POP, POP3, or IMAP.
The type in which you can check your mail from any computer with a browser (what you're using right now) is called "Webmail". Examples of this include Gmail, Hotmail, and Yahoo Mail.
You're almost certainly going to want to be able to receive email at your cool new domain, and if you are planning on checking your email when you're on the road, but don't have a laptop, then you may want to get a package with Webmail included.