I'm a beginner at Ancient Greek as well, and I'm currently using Athenaze, by Maurice Balme and Gilbert Lawall.
I haven't had much experience with other textbooks, but this one is working well for me. The book teaches using stories written in Ancient Greek. There are 16 chapters in the book, each chapter having 2 stories -- meaning that there are 32 stories in total.
Each story has vocabulary words introduced, with the necessary forms given. The stories mainly revolve around the life of a Greek farmer and his family, with some historical stories mixed in as well, such as the story of Odysseus and the Cyclops.
Between each chapter are exercises to help you understand new concepts that were introduced, such as new forms, voices, tenses, etc. These exercises include sentence translations, modifying verbs, and other things. As new forms are introduced, any necessary endings are given. Irregulars are constantly shown, such as the word "to be". You can also find essays about ancient Greek history between each chapter, which may be worthwhile to read if you are interested.
The back of the book has charts of noun declensions, verb forms, and irregular verbs. There is also a glossary from Greek -> English and from English -> Greek. The words in the glossary are mostly the ones that are found in the stories. After that, there is an index that lists where history-related things are mentioned in the book.
I hope this has helped, and good luck on your studies of Ancient Greek.