All these terms, as you correctly state, are nouns.
produce: As a noun, a "produce" is a foodstuff gathered from a harvest, crops, or any other form of food manufacturing. Other synonyms may include "food", or "yield." It is the end result of a production, a harvest, or a crop.
In a sentence you can say:
- The supermarket has a produce section that sells fresh vegetables.
- Look in the produce section for organic greens.
production: A production is a creation, a construction, an invention, or a fabrication. It is something someone puts together for an end result. It is the end result of a creative process.
- The latest science fiction movie is a productionby Steven Spielgberg.
- This production is sponsored by the American Medical Association.
Your last noun is product:
A product is also the end result of a production, or process, but it is semantically associated with items that are manufactured for specific purposes. A product can also be the end result of a multiplication equation. However, in terms of items, similar terms to "product" include: an artifact, an item, an object, a produce (like from before), or an invention, an outcome, a creation, and an end result.
In a sentence:
Cheese and ice cream are products of milk.
The clothing industry manufactures a myriad of new and innovative products for fashionistas.
I hope this helps.
Always remember to use the thesaurus for help with terms that you cannot connect.