I recommend not "investing" in a city-state unless you can get some influence for free to start out. Killing Barbarians or doing another quest for them will get your foot in the door, and let the $250 you have to spend periodically give you a lot more value. For instance, that $250 will be keeping you above the "Allied" line instead of just barely making you friends with the city-state.
Likewise, as mentioned above, the Patronage tree will give you significant bonuses to your influence over city-states.
It seems that the design is such that in the early game you won't be able to guarantee a city-state ally without committing a lot of cash.
To elaborate (not 100% sure on all these numbers but the overall point is still valid):
If you spend $250 to become Friends with a city-state, starting from 0 influence, you'll get 35 influence, making you barely Friends (30 influence). This means you'll be friends for 5 turns, at which point you'll need to spend another $250 to be friends again. You probably don't have another $250 that soon, though, so the influence runs all the way down close to zero before you're able to spend another $250, which again only gives you a few turns of friendship. This is obviously unsustainable.
If instead you perform a mission for the city-state and get 30 or 40 influence to start with, every time you spend $250 you'll get to use all 35 points of that influence before needing to spend another $250.
Similarly, you'd be better off saving your money to start with $500 worth of influence than buying $250 and not being able to afford more right away.
Long story short: if you're already friends, you get more value for your money. Likewise, if you're already allies, you get even MORE value for money you spend to sustain at that level.
You can turn on the "resource icon" overlay which will put up floating icons showing what all the resources are (Ctrl+R, or click on the "toggle map options" button to the left of the mini-map and tick "resource icons").
If you then zoom out you should be able to see where the resources are quite easily, and what they are.
Best Answer
I've had a crack at implementing my first Civ mod to solve this problem: http://steamcommunity.com/sharedfiles/filedetails/?id=169154575
Currently a bit of quick and dirty hack, but it works. It adds another menu item "Antiquity Sites Only", which as you can here means that only Antiquity sites resource icons are shown.
Source code: https://github.com/therefromhere/Civ5AntiquityIconsMod