やること

暗黙的 Intent を飛ばして音楽ファイルの Uri を取ってくる。
Uri を元に音楽を MediaPlayer で再生する。

プログラム

レイアウト

activity_main.xml

<?xml version="1.0" encoding="utf-8"?>
<android.support.constraint.ConstraintLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    tools:context=".MainActivity">

    <TextView
        android:id="@+id/textView"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="this is music picker test"
        app:layout_constraintBottom_toBottomOf="parent"
        app:layout_constraintLeft_toLeftOf="parent"
        app:layout_constraintRight_toRightOf="parent"
        app:layout_constraintTop_toTopOf="parent"
        app:layout_constraintVertical_bias="0.093" />

    <Button
        android:id="@+id/button_register"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_marginStart="8dp"
        android:layout_marginTop="52dp"
        android:layout_marginEnd="8dp"
        android:text="register"
        app:layout_constraintEnd_toEndOf="parent"
        app:layout_constraintStart_toStartOf="parent"
        app:layout_constraintTop_toBottomOf="@+id/textView" />

    <Button
        android:id="@+id/button_play"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_marginStart="8dp"
        android:layout_marginTop="44dp"
        android:layout_marginEnd="8dp"
        android:text="play"
        app:layout_constraintEnd_toEndOf="parent"
        app:layout_constraintStart_toStartOf="parent"
        app:layout_constraintTop_toBottomOf="@+id/button_register" />

</android.support.constraint.ConstraintLayout>

Activity

MainActivity.java

public class MainActivity extends AppCompatActivity {
    final int MUSIC_REQUEST = 100;
    Button buttonPlay;
    Button buttonRegister;

    Uri audioUri;
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        init();
    }

    private void init() {
        buttonPlay = findViewById(R.id.button_play);
        buttonRegister = findViewById(R.id.button_register);

        // Set Listener
        buttonPlay.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View view) {
                play();
            }
        });

        buttonRegister.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View view) {
                register();
            }
        });

    }

    private void play() {
        MediaPlayer mediaPlayer = new MediaPlayer();
        mediaPlayer.setAudioStreamType(AudioManager.STREAM_MUSIC);
        try {
            mediaPlayer.setDataSource(getApplicationContext(), audioUri);
            mediaPlayer.prepare();
        } catch (IOException e) {
            e.printStackTrace();
        }
        mediaPlayer.start();
    }

    private void register() {
        Intent pickAudioIntent = new Intent(Intent.ACTION_PICK,
                android.provider.MediaStore.Audio.Media.EXTERNAL_CONTENT_URI);
        startActivityForResult(pickAudioIntent, MUSIC_REQUEST);
    }

    @Override
    protected void onActivityResult(int requestCode, int resultCode, @Nullable Intent intent) {
        super.onActivityResult(requestCode, resultCode, intent);

        if (requestCode == MUSIC_REQUEST && resultCode == RESULT_OK) {
            audioUri = intent.getData();
        }
    }
}

参考