Why do I get that funny stomach drop feeling right after takeoff?
It happens when the airplane levels off after takeoff, usually either at the first assigned altitude or at a safe altitude where it will be accelerated in order to retract the flaps.   The feeling is a result of negative vertical acceleration.  It can also happen at other times during the flight, such as during turbulence or when starting a descent.

Basically you and the airplane are both in a steady climb and when the aircraft levels off you continue to move up faster so it (via the seatbelt) keeps you at its same level and decelerates your own personal vertical climb.

It is just like the feeling you get in a roller coaster.

My wife calls it a wee-hill!  :-)



EDIT:

Another more familiar example happens anytime that you are traveling in a car:

If you are in a car moving at a constant speed (let's say 55 MPH) you are moving at the same speed along with it.  

When the brakes of the car are engaged, they slow the car down but your inertia tends to keep you moving in the direction that the car was going.  This will be more noticeable as the brakes are applied harder and harder because the deceleration of the car is more extreme.  

You will notice this as pressure against the seat belt as the car holds you in place.  You are still moving forward, but you are getting slower (decelerating).  In an airplane the exact same thing is happening only it is in a vertical direction instead of a horizontal direction like in the car.

What you are feeling is your stomach floating up a little inside your body, because while the seat belt is holding you down, it doesn't hold your stomach in place!