Farming began very early in Rome, and because war made farms abandoned, farms were used less. As soon as the Roman Empire began, farms were in a great abundance. Very early on, Romans realized that their herds' waste would enrich the soil. They had an experiment to see which fertilizer was the best, and it turned out that one of the best fertilizers were a chicken's waste. Thus, we can say that the, "Modern fertilizer industry" was born from this experiment. People, who did not have herds, used house compost, like eggshells and ashes. The Romans also knew what crop rotation was. They planted beans and Alpha in alternate years. This was to replenish the soil.
Mining was very important in Rome. They learned from the Etruscans for where the rich mines were, and from the rich mine they derived a lot of minerals. They used purified tin and copper to make bronze; which was used in weapons. They also mixed copper and zinc to make brass; which was used for very fancy armour. But their most important thing that they discovered was how to harden iron into steel. This would give the Romans and new weapon that no one else had, the steel sword. Steel would not break in wars, and so it was a very valuable discovery.
Farming and Mining both were used for very important purposes: farming for food, and mining for war.