Mastodon for Android (official): Google Play, source Only recommended for beginners. Note that this app won't show the federated timeline. The local timeline can be found under 'explore > community'.
Mastodon for iOS (official): App Store, source Only recommended for beginners. Note that this app won't show the federated timeline, has no option for the Quiet public post privacy and you can't use our notorious maximum of 1312 characters. The local timeline can be found under 'explore > community'.
Although not recommended by us, because these apps are not open-source, other people recommend these iOS apps: