There are special resource tiles on the map that provide food when you build an improvement on them. You should always start by a Fertile Land tile, but there are others you can find (Wheat, Pumpkin patches, Oasis, etc). Build your first city right next to one, then click the food tile and there will be a "Build" button on the actions UI bar to construct the harvesting improvement. This will give you food for housing. Food doesn't accumulate, it's more of "this is how much you can support", so once it's gone you have to search out more food tiles to keep your cities growing. Caravans (researched from Diplomacy still, I think? Should be the Trade tech) give a food bonus for the cities they trade with.

As your cities grow and your zone of influence expands, you can build any resource improvement (Gold mine, crystal mine, iron mine, etc) within your zone of influence the same way you built the farm.