What's a guy got to do to get some food around here?

Either find more resource tiles that give you food (there's like 3-4 different), or build improvements to boost food production, or make caravans build roads between that city and your other cities. I also believe there's a repeatable research in civilization that adds 10% each time it is researched, but I could be wrong there.

Edit: Or cast the spell nature's bounty on each of your cities.

You can also research one of the paths of adventuring to unlock other food sources. Food is supposed to be scarce though, you're pretty lucky if you can make a level 5 city without several others to support it with food.

I have been building Pioneers to make other cities and getting caravans between them. But I don't know if that helps since I got my third city up last night before going to bed. But when I see a special resource I build a city next to it, if it is away from another city.

  Hey, by doing multiple research in the "adventure" line, there is an option which reveals more resources of different sorts each time you choose it.  It usually makes bees and other types of food pop up all over.  I have no problem for foods after having combined that along with researching irrigation/granary/food guild from civic.

I've seen some discussion that Empire factions don't get as much food as Kingdom ones.

If you click on a caravan once their route is established, you can see the food bonus theyre currently providing.

There's more tech available to boost food production than almost any other resource it seems, so do don't neglect your civics research.

Don't establish cities willy nilly.  Have a good reason for doing so, A single resource tile is probably not a good enough reason to found a city, unless it is a vital resource that you have to have right then.  There should be several attractive resource tiles in the eventual influence of a new city, or it's probably not worth the tradeoffs from founding a new city. 

I'm playing Empire and I don't see any of these researches. I've found all of one food resource tile in a huge swathe of map. I can't expand my territory because I'm out of food as well.

It's strange - I've never had a problem with food in the 4 games I've played fairly extensively.  Like someone mentioned, there's a research path in the adventure tree that unlocks new resources.  I believe the third in that tree unlocks food resources.  Once you do that, and once you start building things like irrigation and other food bonuses, you'll have TONS of food.  I've never been limited in how much I can grow any of my cities (even when I had 30+ cities) because of lack of food.

Try playing Empire where you don't have any of those tech trees.

Caravans will help the city that builds them with a percentage food bonus. It does not help cities that don't make food.

I was hurting for food in my last game when all of a sudden I had extra food out of nowhere. Hadn't built any new improvements or anything. So I build more housing to level my city and later on I was horrified to find out my city has -3 prestige because of a food shortage Turns out all the extra food I got was from caravans from other neighbouring nations and when they slowly started being conquered by the AI the caravans stopped and I had a lack of food. I wish there was a chart that showed me what resources were produced from where so I don't commit to making a large city just to find it being deserted several turns later

I'm still hazy on how caravans work atm. Maybe once I figure that out I'll do better on food and gildar trade

Well that's wrong because I'm gettin food all over the place. I found Oasis's and fertile lands all around me and I'm researching things that increase food production by 25% like granaries and other neat little buildings. Food can be plentiful but you aren't going to get it all in 10 turns. It takes a lot of turns to get going good. Don't forger your research buildings and those lost libraries and stuff out there that really give you huge boosts in research. Soon you'll be making discoveries every other turn or so.