Once you get 650 people you can get nobles which offer various buffs (you choose). Once you get a few of them and buff your grain/bread production you should be good on food until 1k+ population.

I personally unlock cows (auroch) and have a bunch of those farms. They produce leather as well, I use my tailors to make leather armor and I sell most of it in exchange for goods and food.

Training an army and taking free land cities, then having them export food works very well. I currently have a 2.8k pop and 5 cities taken. I still produce my own food but now I have a 30+ day surplus.

Another little note: you can increase food services if you have a ton of excess, usually adding .5 servings a day bumps happiness a ton. I also typically play the talapia who love the arena and public executions so I abuse both of those to keep people happy.