If hot air expands in all directions why is the balloon moving up? A balloon is moving up. The hot air inside the balloon is expanding in all directions and cold air outside the balloon acts with force on the balloon from all directions (higher pressure force). Why is the balloon moving up then?

 A: First and foremost, a primer over what "buoyancy" is is needed. Pressure decreases with altitude. The atmospheric pressure at the top of the balloon is a tiny bit less than the atmospheric pressure at the bottom of the balloon. This pressure difference results in a tiny net upward force on the balloon. The balloon rises if this tiny net upward force exceeds the tiny weight of the balloon.
The balloon doesn't always move up. Try releasing a hot air balloon in Los Angeles or Denver or Mexico City. Oftentimes the balloon will rise maybe a thousand feet and then just stop. This indicates the atmosphere is stable, and it's why Los Angeles, Denver, and Mexico City have such serious problems with smog.
A hot air balloon rises when the environmental lapse rate is sufficiently high. Air is a relatively lousy conductor of heat. This means the balloon doesn't transfer much heat with the external environment. The conditions inside the balloon more or less change adiabatically. Assuming the balloon is filled with air (as opposed to helium or hydrogen), the balloon will be buoyed upward so long as the temperature inside the balloon remains higher than the temperature of the surrounding air. The balloon stops rising shortly after the lapse rate of the surrounding air is less than that of the adiabatic drop that's happening inside the balloon. 
Air temperature tends to decrease with increasing altitude. That "tends to" is key. The temperature profile eventually flattens out. Sometimes the altitude at which this happens is dozens of kilometer up, at very top of the troposphere. The boundary between the top of the troposphere and bottom of the stratosphere is where the tropospheric lapse rate stops. Temperatures rise with increasing altitude in the stratosphere. This is the key reason weather generally doesn't reach into the stratosphere. It takes a very large and very powerful thunderstorm to punch through the tropopause, the boundary that marks the top of the troposphere / bottom of the stratosphere.
Other times, this transition to a stable temperature happens at much lower altitudes (e.g., Los Angeles, Denver, and Mexico City). 
