Most movement of particles (such as air) happens due to diffusion. If you put sugar or salt into water, it will eventually dissolve into the whole volume of water. In this example, it's because there is originally a different concentration at one point than another (that is, there's a lot of sugar sitting on the bottom of the cup, and none at the top). This creates a gradient, a difference in concentration in this case, which can do work (like move particles!).

Hot air rising happens for similar reasons. Hot air is less dense than cold air. When the air gets hot, its density decreases, and now there's more air molecules in the colder zone than in the hot zone. There are more complicated things happening, but for this simple reason alone, we might expect hot air to rise for the same reason that objects less dense than water will float in water.