From language learning I know and enjoy the spaced repetition learning flashcard app Anki. The desktop software is free and open source. For phones there is an official iphone app (costs money) and a free open source android app created by volunteers, but you can also do revisions by using the mobile friendly website without any app. Spaced repetition learning is very powerful as the algorithm decides for you when you should do your revisions, in increasing intervals if you answer correctly.
Anyway, there are several user-created bird decks. I only studied those for western Europe so can't give a specific US recommendation but you can find bird decks here: https://ankiweb.net/shared/decks/Birds
For just learning US bird calls e.g. the deck "Birds of North America" might be sufficient. Or use the Ultimate Birds deck which is huge and has birds from all over the world. With the Ultimate Birds deck you can use tags to create a filtered deck which only contains species of North America (the posted link contains instructions). The notes of the deck contain information in fields that you can use to create different types of cards via card templates. Fields are e.g. sounds, images, English (English name) or Scientific (scientific name). For call recognition you can create a card type sounds -> English which has the sounds on the front and the English name and image on the back side. But you can also create card types for recognizing images or learning scientific names etc.
Anki has a learning curve, but there's lots of material online about Anki, e.g. tons of youtube videos (apparently many med students use it for memorization) and it can be very powerful.