In season one, Daenerys meets Drogo by being paraded in front of him when he comes to Magister Illyrio’s palace. The exchange is quick and they don’t share any words. In the book "A Game of Thrones," Illyrio still brokers the deal like in the show, but Dany goes to a party at Khal Drogo’s Pentos palace to meet him there.
(Yes, Drogo has a palace in Pentos). Also, Dany has already met Ser Jorah at this point in the book, and he accompanies her to the palace party, whereas in season one she meets Jorah at her wedding.