Level 1- The Streets
Guard
Inside the first building, you'll find a big guard blocking a doorway, seemingly immovable.  Enter the second building and find the coin-flipping bookie that is leaning against a wall.  You can switch clothes with him by jumping up and down.  Now you will be able to pass by the guard and rescue the girl!

The Sewers
Despite their appearance, the sewers are linear instead of being designed as mazes.  Aside from the usual tricky jumps, the only thing to watch for are the gate switches.  These allow Harry to progress further into the tunnels, and consist of two different designs- a large box with a red switch and two smaller, connected boxes that also have a red switch.  No puzzles to solve here, just push up on each one as you encounter it.

Gruff
Stand by the barrels, and time your shots so that you hit Gruff as he lands from his jumps.  It will take 8 hits to defeat him.

Striker
Climb the broken ladder and stand on the rooftop.  Striker won't shoot up at you, and you can easily aim your shots.  It will take 10 hits to defeat him.

The Toymaker
The Toymaker is protected by an invincible creation of his, and defeating him will require Harry to get a little creative himself.  First, shoot the beach ball that is floating in the water.  The easiest way to hit it is to step off of the upper left ledge, land on the lower ledge, and aim down.  Now make your way to the water valve in the middle of the screen and shut it off by pressing up.  Go back and climb to the top of the ladder and wait for The Toymaker to be electrocuted.  Now get on the Toy Boat and ride it to the exit tunnels.

Level 2- The Waterfront
Roll out the Barrel
On the Waterfront level, you will need to use a barrel cross a section of pipe that is guarded by an armed dockworker.  First, kick the barrel into the water.  When it stops at the dockworker's feet, use your Magnum to explode it.

Power Boat
The Power Boat will be a source of constant frustration- unless you know the secret:  Shoot the rope that holds the first moving crate, timing it so that it falls directly onto the boat.

Longshore
They don't call him "Dirty" Harry for nothin'.  You need to repeatedly kick Longshore in the crotch, then hit him with the harpoon gun while he's stunned.  If you try shooting him without stunning him first, he'll simply catch the harpoon and snap it in half.  5 harpoons will defeat him.

Level 3- Alcatraz
The Prison
Avoid the doorways here, they will only return you to earlier points in the level.   Alcatraz is a linear level.                                  

Scaling the Cliffs
Use the grappling hooks to cross the gaps between rocks.  Watch carefully to learn which rocks are safe.

General Ripper
The General will leap over you if you try to shoot or approach him.   Force him to leap by walking towards him, then turn and hit him as soon as he lands.  It will take 8 hits from your Magnum to defeat him.

The Private Helicopter
It will take 16 hits to take down the Anaconda's helicopter.  Avoid excessive misses by keeping your Magnum aimed upwards, then firing when the helicopter is hovering directly above.

The Anaconda
Use the LAWS rocket to defeat the Anaconda.  4 hits will win the game.

Miscellaneous
Necessary Kills
There are some punks that you must kill, as they carry items you need to progress.
1.  The dockworker with the harpoon gun. (he's on the top ledge, after the first tall ladder)
2.  The soldier with the grappling hooks. (right before you come to the cliffs)
3.  The soldier with the LAWS rocket. (on the lighthouse)

The Ha-Ha Room
No, it's not a club, but a hidden room that you might want to check out just for fun.  When you reach the entrance to Striker's alley on the Streets level, don't turn into it.  Instead, continue left to the very last door at the end of the street.   Now enter the very last room in the building.  You will be trapped inside with graffiti that reads "Ha-Ha-Ha", written on the wall behind you.