how do i add an email address to an alert

Do you mean the email address which you want to receive the alert? If so, just add it to the To Other Emails textbox (separate multiple emails by commas). See this article for help

I am new to Marketo - and I want to send out an email blast, but want to be able to receive an email as confirmation that the mailing has gone out.  Is an alert the best way to do it? or do you automatically receive an email once you send out the campaign?

No problem - we're all new at some stage! Are you using the Email Program, or just a Default Program?

There's no alert you'll receive when a campaign has been sent, and you could set up an alert yourself, but at first I would recommend checking the results in the Marketo interface so you're familiar with it all. Generally, we will first send out our emails to a static seed-list only (add as a filter in Smart List tab), as a final testing step. This list will contain leads that are actually members of your internal team. If all is well, you can then remove the static list filter from your Smart List, double check the leads #, then you're ready to go. 

Once the campaign has been sent, you can check the Dashboard (if using Email Program) or Results tab (if using Default Program) to see the status of your campaign instantly.

Don't forget to test thoroughly and take small steps at a time. Hope that helps!