I am new to this. I have a village with 15 villagers all of working age. I have the first two levels of every tech except fertility and sprituality. My problem is it takes so long for my field to regenerate, that I am running out of food.

Even if I put all 15 villagers on farming, I cannot produce enough food. I have 32000 tech, so I am going for the thrd level of farming, but it there any was to supply a village of 15 without the third level? Is there another food source I have missed?

A population of 15 villagers is really more than those two food sources can comfortably support. The amount of food available from both the farm and the berry bush is limited, and, once it's depleted, there will be no more available until it regrows.

Your villagers probably won't starve, though, because they can survive a short period famine. To make life easier for them, you really need to research that last level of harvesting so that your villagers will have an unlimited food source.