Why was "a drink" used? Can I use "drinks" instead?
"Get a drink" is a common phrase used as a shorthand to signify the desire to be with someone in a social setting. You are correct that you could say "let's get drinks" and the overall meaning would be the same. However, people more commonly say "a drink", to imply that there is no obligation to have more than one drink if you don't want to. This is because the "drink" is usually alcoholic. But as I said both "get a drink" and "get drinks" are correct and accepted in this usage.