Sure. Here you are. Feel free to come back to me if you want me to clarify further one or the other point.

First you are correct: heat rises. If you take this into account, this would mean that the atmosphere will become hotter and hotter with the altitude. However (yes there is a catch), this is not the only factor into the game.

Another factor is that gas get cooler when the pressure drops (basic thermodynamics). So, a bunch of air close to the ground as a huge amount of gas pushing on it (i.e. the atmosphere) so that the pressure is large. When the gas rises in altitude, a smaller and smaller fraction of the atmosphere pushes on it, so that the pressure drops. And with the drop of pressure, the temperature drops too.

However, all of these are by far not the dominant factor. Earth itself is. The planet absorbs heat from the Sun and radiates it back at the surface. The higher you are, the less important this effect is. As a result, air is warmer close to the surface of the planet.

PS: I am no nerd... ;)