UPDATE The formula has changed as of the March 2011 patch, it is now
(city population * 1.1) + (capital population * .15) - 1 for each city connected to the capital, not including the capital.
- The Machu Pichu wonder increases the modifier by 20%, to
(city population * 1.3) + (capital population * .15) - 1.
- Arabia increases the constant by 1, to
(city population * 1.1) + (capital population * .15) (thanks WillfulWizard).
The economic overview also gives a far better explanation of how it is calculated - in other words, this question is now trivially solved by simply looking at that economic overview :)
Original answer below.
Okay, I did a bit more testing, and bwarner's answer is almost accurate:
Each city, excluding the capital, provides (city's population * 1.25) + 0.01 gold per turn. Owning the Machu Pichu wonder increases the modifier by 20%, to (population * 1.5) + 0.01, for all the cities.
The capital does not provide any gold.
The 1.25 and 0.01 can be seen in the Assets\Gameplay\XML\GlobalDefines.xml file:
<Row Name="TRADE_ROUTE_BASE_GOLD">
<Value>1</Value>
</Row>
<Row Name="TRADE_ROUTE_CAPITAL_POP_GOLD_MULTIPLIER">
<Value>0</Value>
</Row>
<Row Name="TRADE_ROUTE_CITY_POP_GOLD_MULTIPLIER">
<Value>125</Value>
</Row>
The economic overview also demonstrates these values.
I couldn't find any other factor which affects these values, and I checked different difficulty levels, different distances between cities, different city route type (road vs railroad, road/railroad vs harbor), different city health and whether it is occupied. Looks like it's solely the population.
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.
Best Answer
There are two basic reasons to connect your cities by roads: for Trade Route income, and for Unit movement
Connecting for Trade Routes
Warning! These calculations are now suspect because of new info found about how road maintenance works. I will update this with any changes needed once we have verified numbers for road maintenance costs.
Summary: only build trade routes to cities with 6 population or more. If you care for the math, then continue reading...
Now that Oak has solved the Trade Route formula for us, we know that normal trade income is very close to 1.25 * population of the connected city. (not the capital)
So, suppose you have your capital and one city, they are separated by 6 hexes. You'll be paying to maintain each of those roads, costing you 6g. So, at a population of 3, you would make 3.75g, be spending 6g, and be losing money overall. Your worker time is EASILY spent doing something else. Of course, if your cities are closer together, the calculations look better. Finally, I believe that Harbors cost 2g to maintain. Realizing that you only pay for your capital harbor once, sea routes effectively only cost 2g to maintain.
I would like to propose a rule of thumb. You can pick whichever population number seems best to you, but I believe my worker can do better than even a 4g improvement by working other tiles in the time it would have taken him to build that road. Thus:
Only build trade routes to cities with 6 population or more.
If you follow this rule of thumb, you will, at the least, never lose money on a trade route. Further, if you start working on the trade route when the city is size 6 already, there's a good chance it will have grown to 7 at least by the time you finish, and you can actually start making enough money for that worker time to be worth it.
Final note on trade routes: Arabia gets a flat +1 to trade route income. (AFTER the multiplier, so simply add 1 to every gold number on the chart.) So feel free connect Arabian cities at population 5 or more, if you like.
Connecting for Unit Movement
We don't have nearly as good of data on this, but from my Civilization experiences, roads are VERY worth building if you are producing units and sending them to war. You get your best units at the front for longer before they are obsolete, and you'll have more units at the front at any given time.
If you're not moving units for war, roads may also be worth building to quickly move defenders in the event of being attacked, by barbarians or another Civ. I think you can just ask yourself:
If any of those it no, the road is not worth it, cause it won't help.
And if you're not at war, and not defending, I doubt you want to build roads just to move units, because you don't sound like you're moving units around much at all!