This game has TONS of secret rooms. Here are some of the most lucrative and useful.
Shortcut through the Hall of Chains - Don't bother trying to jump your way through this difficult area. Upon entering, leap up and climb the chain directly overhead. You'll go partially through the ceiling. Jump around and play with the controller until you disappear into the Golden Chain. Grab the goodies and fall down the hole at the end to find yourself at the end of the Hall.
In the lower lefthand corner of the Front Hall, press up to enter Pugsley's Den. Take the goodies, stand in the upper lefthand corner of said area, and press up again to reach a large hidden area with cash galore, weapons, and 20 1-Ups! Outside the house, find a fez near the gallows, fly around the house, and drop in the chimneys for more cash and prizes. Check 'em both.
In the Armoury, look for a hole in the ceiling near the end and a switch one screen to the right that produces a large square of blocks. Get on the blocks and jump off high and to the left. You'll disappear into the Trophy Room, where you can collect mucho goodies. Also crawl into a bear's mouth in the Lower Gallery and check out the Closet at the end of the Weaponry. (Go get a fez from Fly, Fly, Fly and return to soar up to the Closet.) In the Library, use the "A" block in the very beginning as a stepping stone to the Top Shelf in the lefthand wall. Press up in front of the "door" book in the Reading Room to reach a Secret Panel with power-ups. In the Big Books area, take a sword, press down while standing on the "drop in" book, and use the sword inside to catch a 1-Up.
There's lots to find in the conservatory scenes. In the Conservatory, press down on a plant pot to find lucre inside. Walk left into a certain column in the same area (there's a shoe floating to its right) for cash in the Privet. Walk right into a wall in Poison Ivy to collect goodies in a Secret Garden.
On your way to Pugsley, try pressing up when standing on doorjambs. There are also hidden "ropes" between some of the Toy Tower screens. Try jumping into the far corners of the screen or pressing down to find said ropes. After saving the Pugmeister from the Wacky Scientist, stand where he was held captive and press down to find $$$ in the aptly named Pugsley's Pipe.
To find more secret stuff, jump down every single bottomless pit you see. Every one of them contains something good. This is one of the few games that won't kill you instantly for touching something ouchy or accidentally falling into a hole, so take advantage of it! In The Addams Family, you're actually rewarded for goofing around.