Roads are expensive in Civ 5 (1 gold upkeep per turn), but I make some of that back when I connect my cities to my capital. Is it always still a net loss, and if so, is that money better spent elsewhere early in the game?
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!
It depends how close the cities are and what their populations are. If you look a the related question about the trade route income formula you can calculate whether the roads will be profitable or not. If the cities are close, then the maintenance cost of the roads will be less than if the cities are very far away. If the cities are large, the trade route income will be very good. If the trade route income is greater than the road upkeep, then it's a win, at least financially.
You also have to consider the opportunity cost of the workers building the roads. If they didn't spend those turns building roads, what would they have built instead? Would that other thing they build have been more beneficial than the profits from the roads?
Also, think about the usefulness of the roads themselves. Having roads greatly increases the speed with which you can move units. If you have one city that is very good at producing military units, but is far away from where the battle is, you are really going to wish you had those roads, even if they cost you money.
Lastly, consider roads to city states. Any nearby city states will eventually offer a quest to be connected to your capital by roads. Completing the quest will make them very happy with you, and that could result in great rewards compared to the cost of the roads. For example, a military city state as an ally will give you multiple units, definitely worth a few gold per turn.
You also should consider the value of the Meritocracy Social Policy in the Liberty branch:
If you are going for a hybrid science/domination win, it can be very helpful.