Castlevania: The Adventure isn't much like the console titles, both in game mechanics and in the method of secret-finding. Instead, you can find secret rooms in Adventure by maneuvering your way into sealed-off areas of a stage by manipulating certain conditions--and there's one such instance of this per stage. Below are the descriptions on how to reach each stage's secret room.
This one isn't too far into the stage. After climbing up a rope, you'll be in the wasteland area of the graveyard, where goblins patrol the area. While on the entry rope, just continue to climb straight upward; you'll pass right through the bricks and into the room seen top-right. There you'll find an energy restore, a whip power-up, and a 1-up symbol--a combo you'll find in all such secret areas.
After crossing a long bridge, where rolling eyes drop down and roll all around, you'll enter into the room seen top-left. Stand where Christopher is placed in the the screenshot and wait for a rolling eye to roll onto the third step down. When you strike the rolling eye at this point, it'll explode and leave a gap in the step. Jump into the gap and slide down the invisible rope to reach the secret room.
On this insane stage, the secret room is located right near its end; that is, it's mainly relief after you've endured all of the madness. You'll climb up a rope into the room seen top-left. Jump off of the rope and to the left, where you see Christopher in the screenshot. You'll walk right through the wall and into yet another secret room.
Somewhere deep into the final stage of the game, you'll find yourself in a room guarded by a single Zeldo (the room seen top-left). After finishing it off, climb the rightmost rope until you reach just below the ceiling and jump off to the left; you'll land on an invisible platform that stretches out to the other side of the room. Walk to the halfway point of the room and jump up to cling onto an invisible rope. Climb it to enter into the final secret room.
Early Life: It's no secret that Adventure is a very difficult game, both in design and through frustrating controls, and that you'll need every advantage. So let's start out with a boost: On the first stage, avoid whipping the first ten candles you come across. When you arrive at the eleventh candle (it's right about where the Rolling Eyes first start appearing), whip it to receive a 1-Up that ordinarily would not have been there.