Here is my understanding of the boiling process: pressure of the atmosphere is pressing down on the liquid and this pressure is propagated throughout the liquid. That is, pressure everywhere inside the liquid is equal to the ambient pressure (I'm ignoring the weight of the liquid columns for simplicity). In order for bubbles to form inside a liquid, the gas molecules must push outwards with a greater pressure than the existing ambient pressure to "make room" for the bubble. The pressure that the bubbles are able to exert on its liquid surroundings is the vapor pressure and hence we need to be at a temperature where vapor pressure at least equals the ambient pressure.
But my source of confusion is that ambient pressure is not the only pressure pressing down on the liquid. There is also the vapor pressure of the liquid in addition to the ambient pressure. Then according to my understanding of how boiling works, this means that the pressure inside the bubble (which equals the vapor pressure of the liquid) must be greater than the ambient pressure plus the vapor pressure of the liquid which makes no sense. So I'm assuming I have an incorrect understanding of how boiling works and was wondering if someone could help me correct it.
But my source of confusion is that ambient pressure is not the only pressure pressing down on the liquid.
Pressure does not press down. When something is under pressure, it exerts it in all directions. Maybe it would help imagining this problem in the absence of gravity (you would have to put the sample in a stretchy balloon to get some pressure while being able to change the volume).
So the balloon would exert some pressure on the liquid. If the liquid is hot enough that the vapor pressure is higher than the pressure, and if you can get some tiny bubbles going, they will expand as more and more liquid goes into the vapor phase, increasing the volume of the bubbles while maintaining sufficient pressure not to collapse.