Continue game:
While falling, hold I and press Run.
Princess mode:
At the title screen, hold I for about 10 seconds.
Secret Test Mode:
At the title screen, hold down Run while pressing Select six times, then release. Hold down I, II, Up and Left, then press Run. Press Select and Run simultaneously to reach each part of the Secret Test Mode. You can access a Sound Test, select your starting stage and your starting number of ships, among other things.