Applicable to all versions based on the Sega Genesis one: Genesis, Atari Jaguar - without the extra levels. Sega CD, Sega 32x, Windows PC - with the extra 3 levels.
See below the embedded videos for the quick rundown of tech.
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Tech
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For rapid fire, hold Jump and mash Fire whilst standing.
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To let go of vertical ropes quicker, hold Up just before grabbing them.
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To cancel fall recovery, let go of Left/Right before landing and hold either Fire or Whip.
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To turn around without skidding, jump forwards and turn around in mid-air.
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Buffer your inputs, this makes Harry perform actions at the first possible frame.
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Do NOT change the default controls. This will either softlock the game on the minecart level (on Sega Genesis and Sega CD) or disable shooting using the X, Y, and Z buttons (on Sega Genesis, CD, and 32x).