I have fruit trees and I used to have a large garden and what I have noticed, is that they don't seem to give me long term yields. Mostly it's about a week of ripeness, before they go bad, or the birds eat them. And the problem is that if I pick them all before they go bad, they usually don't last long in my kitchen. How would ancient people's maximize the effectiveness of their crops? How could fruits and vegetables stay edible to last them through the winter?
I'm not sure this is quite the answer you are looking for, but one of the key methods used by ancient civilizations for storing calories was utilizing the liquid forms of their crops, alcohol.
While today we think of alcohol as a complement to social activities and leisure, it served an extremely important purpose throughout history including ancient history. As mentioned in your post, before refrigeration and modern day preservation and agricultural methods there were no reliable ways to preserve and store crops in their solid form.
But by turning it into alcohol, you could have a reliable source of calories that was much easier to store and less prone to going bad or being eaten by rodents, insects, ect. to supplement whatever sources of food you had depending on the season.
Almost every major ancient civilization utilized alcohol as far as we know, the Chinese fermented rice, honey, and fruits. The Greeks drank wine and mead, as well as the Iranians, and Egyptians who had multiple variations of beer as well. The indus valley civilizations made alcohol from sugar cane, and other fruits. American civilizations also made their own liquors from maize, honey, pineapples, and other crops.
McGovern, Patrick E. (2015) "Fermented beverages of pre and proto-historic China".
Dirar, H. (1993), The Indigenous Fermented Foods of the Sudan: A Study in African Food and Nutrition
McGovern, Patrick E. (2003) Ancient Wine: The Search for the Origins of Viniculture
To add to what /u/Dom19 said. Ancients had multiple ways of preserving things like fruit. Alcohol was definitely an important one, but there were others.
The oldest cookbooks like the Roman Apicius include things like jam recipes that can keep preserves for many months before they go bad. Techniques for drying or otherwise preserving fruits were well known. So while grapes may be available only for a few months a year, raisins could be enjoyed year round. The earliest raisins date back to around 2000 BC. For that matter a proper root cellar can preserve a lot of fruits and vegetables for months as a time without any further preservation.
There is a limit on how well these will work. Most fruits and vegetables were seasonal in the ancient world. Plants with long growing and harvesting seasons and stable storage were highly valued. It's one of the reasons onions were so ubiquitous across the Mediterranean and Mesopotamia. They had a long growing season. They could be planted as soon as the hard frost passed, they could be harvested within 60 days. Multiple harvests could be made. And they could be harvested right up to a hard frost. Finally they could be stored and kept for the winter months.
Things like strawberries with short harvest seasons, and which kept poorly, were known, but not intentionally cultivated. They didn't join the garden until the modern era. Instead they remained an occasional wild harvest treat.