What three ways is thermal energy transferred in the atmosphere?

Conduction is the transfer of heat via contact. In the atmosphere that only affects the first few meters of the atmosphere that is in contact with the ground. It is a slow process, but it does also form air masses (the atmosphere over a large area of ice in the arctic will form an arctic air mass over weeks due to conduction).

Advection, is the lateral movement of heat. This is where we use the term fronts. Warm air advection occurs behind a warm front. Cold air advection occurs behind a cold front.

Convection is the vertical movement of heat and it involves uneven heating of the Earth and can also involve water vapor. If there is an area that heats more quickly than surrounding areas (say a black asphalt parking lot), the air above that will heat and expand (temperature is proportional to volume). Since it expands it is going to be more buoyant, causing it to rise. This moves heat upward in the atmosphere.

Convection can also involve water vapor. As a parcel of air rises it cools as the pressure drops. Since it cools the amount of water vapor it can hold also drops. Eventually the temperature reaches the dew point and the water vapor condenses. In the act of condensation, water molecules give up some of their momentum as heat and that transfers heat in the atmosphere.