Am I just being an idiot or can't you take over city states?
Been trying for ages to take over almaty but the health keeps going up a bit
Managed to make it a puppet, no idea how
You're just being an idiot
You can conquer city states, but doing so is even more idiotic
The long bar above city states is their influence level. There are several stages of influence - <0 = mad, 1-30 = neutral, 30-60 = friends, 60+ = allies.
At friend and ally status, the bonuses that city states give you are incrdible, but only from Maritime and Cultural ones. Militaristic ones will give you some units, but they are usually just warriors and scouts, however they can help you out incredibly during a war.
When you go to war with an AI, so do all your allied city states, and the number of units they churn out is brilliant and keeps your enemies pressured. Maritime city states give you good. It is a far more viable tactic to befriend a couple of maritime states to feed all your cities so they can specialise in working production tils instead and completely skip having to build expensive granaries and water mills.
Cultural city states vastly improve the amount of culture you get per turn. Furthermore, the patronage social tree reaps even more benefits from city states - Influence degrades 25% slower, minimum influence of 20, +25% influence from gold gifts, +33% of an allied city states science output gets added to yours, double the happiness from gifted luxury resources from city sates, and finally a chance of allied city states giving you great people - that last one is just awesome.
Patronage is the most powerful social tree to develop, and you should do as many quests for city states as you can (except for killing others).
City states also have personalities. Never waste your gold or time on Hostile city states because influence degrades faster with them. I wouldnt bother with Irrational or neutral militaristic ones either as the bonuses they give arent that great.
Friendly (most preferably) or Neutral Maritime and Cultural city states will make you very powerful if you befriend them from the extra food and culture they provide. If you get several friendly cultural and / or maritime states, developing the Patronage tree is a no brainer, do it and save up 1000 gold as much as you can for the best rewards. Friendly states never request you to capture another state, they usually ask you to clear a barb camp, build a road to them, find an AI, build a wonder, or spawn a great person, and meeting those requests if you can nets you huge influence boosts.