Yes, you can get the discounted phone price with any plan. Go to wireless.att.com and put in your zip code and see what plans are available in your area. As I said, the $59.99 family plan may not be available in all areas. I am on it now, but I read someplace that it depends on the area of the country. I think that $39.99 price on that website you posted was a mistake. Either a typo that was meant to be $59.99 or they mistook a individual plan for a family plan.
$59.99 for two lines, $30 for a data package, plus any text messaging or other features, plus 15% taxes and other charges. If you want unlimited texting, it is $20 a month per line, or $30 for all lines on the family plan. They also offer 200 texts for $5 or 1500 for $15.
You are looking at about $100 a month (with no text messages) after taxes and fees.