Expenses
The top two things that cost you money in Civ 5 are:
To reduce the first one, be strategic about what you build where. Cities should be focused on a task. Pick terrain, tile improvements, and buildings that compliment this focus. For instance, if you are building a production-focused city, build it near hills and forests, and focus on military and production enhancing buildings. Don't build science or cashflow generating buildings, since you won't be producing much of those in this city.
Likewise, if you are building a science focused city, don't build military or other production enhancing buildings there. The city's focus should be on science or economic enhancing buildings instead.
The only exception is happiness - in order to maintain a positive balance on the happiness scale, you'll likely have to build some happiness generation buildings in most or all of your cities. This tends to be one of the largest trade offs economically in most of my games - do I make the people happier, or do I bank some extra gold?
You can also guide each city's governor to focus on a particular area, and this can help your citizens understand what they need to do in order to support your vision for the city. If you leave the governor alone, the city may ignore some of your tile improvements in order to work tiles that create something you're not interested in, but which makes the city's output more balanced.
Also, don't let your workers run free - focus your tile improvements on whatever works best for the city the tile is in the radius of. Automated workers don't always pick the best terrain improvement for you. They may destroy forest or plant farms around a city that is going to be production focused, for instance.
Every unit you produce costs you money every turn as well. Try to focus your attention towards a small, but mobile army, and don't fight on multiple fronts if you can avoid it. Mixed unit tactics involving ranged artillery units and close-up front line fighters work best. The AI is terrible at war, and they will generally start on the offensive, even though you've likely got your army massed and ready to funnel them into a meat grinder.
If you find yourself with a strong surplus of units, gift some to an allied city-state. You'll gain influence with them, and if you declare war, those units are likely to still benefit you indirectly.
Income
You gain money via citizens assigned to money-generating tiles, trade agreements with other Civs, and by trade routes between cities. Any duplicate of a luxury resource does you no good - it is there to trade.
If you end up with 2 of a luxury, or more of a strategic resource than you need, find someone rich and sell it to them for a tidy profit. Suppling your possible enemies with strategic resources might sound like a bad idea, but since you control the flow of that resource, if they start doing something you don't like, you can cut them off and drastically reduce the effectiveness of any resource-requiring units they produced in the meantime.
There's also a glitch that I don't believe has been patched whereby if you destroy a luxury good improvement (thereby reducing the amount of that luxury good available) you can end a trade agreement early at no penalty, and then rebuild the improvement and sell the good back to that same Civ for another sum of money. This could be considered cheating, and I imagine someday it (will be/has been) patched however.
Roads between cities are generally profitable, but you want to try to minimize the number of redundant road links you have, as having 2 roads between 2 cities counts the same as having one, and costs you extra.
There are many reasons to go to war with another Civ:
- Take their territory and resources - although the game is more balanced towards making small Civs profitable than ever before, in most cases having more territory/cities/resources can still give you a big advantage.
- Halt the progress of another Civ who is doing better than you in some other area - if you have the military, you can stop a Civ who is doing better in technology or culture but who is weak militarily.
- Defending allies or forging new alliances - war is a powerful way to increase your reputation with allied civilizations. Allies can make war against you more difficult, and they'll be more receptive to trade agreements that are beneficial towards you.
- Nip a potential adversary in the bud - the AI in particular is terrible about unit tactics, so you may be able to provoke them to attack you and devote resources to war before they're completely ready.
Depending on why you're going to war, your goals/result will differ. You may wish to simply weaken their army, force them to give you money or other resources, or take a strategic city, but if you're going for a Domination victory, you're going to want to capture or destroy all of their cities, despite their (likely) pleas for peace.
Be wary of growing too large too quickly, as happiness is critical to growth, and extended war and capturing cities leads to rampant unhappiness. If you devote too much of your Civ's resources towards building and maintaining an army, you may fall behind in other areas, giving others room to outpace you.
I tend to prefer to puppet captured cities (instead of razing or annexing them) as this tends to provide a pretty good balance between the benefits of having more territory, resources, and population and the detriment of unhappiness due to size. There may be cities that are just worthless (ie, population 1 or 2 cities in poor positions) and need to be razed, or cities that are extra valuable (ie, high population cities with wonders or other beneficial buildings) that you'll want to keep tighter control over.
Unit tactics have also changed significantly since Civ4 - in that game, you wanted to build large stacks of units to march across enemy territory with at least moderate losses. In Civ5, if you focus on a few powerful front-line defensive units coupled with ranged attacks from siege units, you'll find that your investment in units is small compared to enemy losses. Siege units in border cities can make mincemeat of overzealous enemy armies once war is declared.
Best Answer
I think you're hiding 2 questions here: when should you use nukes, and how should you use nukes. But first, I think this question beckons a little more explanation of how to win a game, especially since you mention both nukes and the United Nations.
How to win a Civ 5 game?
There are multiple ways to obtain a victory in Civ 5.
This is not the place to get into details on all the above conditions, there's the manual and other sources for learning about that - but the key is, if it's already late game (and it is if the United Nations is in play), you should already have a strategy in mind.
If you still don't have a strategy for winning then stop, review the above conditions, and decide which one is best for you. The United Nations is already in play, meaning that if one of the civilizations managed to obtain a majority in the vote, the game is over - though the only way to do that, when there are just two civs, is to bribe all the city-states to vote for you. On the other hand, if there's only one other civ in play, a blitz action to conquer its capital is likely to succeed, especially when you're sitting on a stockpile of nukes.
How to use nukes?
In my eyes, nukes can serve two different roles - as a strategic weapon to weaken the enemy's economy, or as a tactical weapon to destroy units and weaken cities before conquering them.
As a strategic weapon, cities getting nuked will lose half their population and some of their non-wonder buildings, and many improvements around them will be destroyed; recovering from that can take a lot of turns. One nuke won't make a lot of difference, but if you have a bunch of them, and you have quality enemy cities in range, you can put a serious dent in your opponent's economy.
If you intend to use nukes as a strategic weapon, I recommend spreading the nukes around (1 or 2 per city), because you want to get the most out of the improvement-destroying and the fallout.
As a tactical weapon, nukes can cause damage to cities, making them easier to capture; but where they truly shine is the damage they inflict to units. A nuke will severely damage all units in range 2, possibly killing them, and this can be a great way to break enemy formations and fortifications. Nukes are also useful for killing units hiding inside cities, especially missiles and planes.
If you intend to use nukes as a tactical weapon, just target clumps of enemy units. Try to force them together - for instance, by creating a threat at a certain location. Don't waste a nuke on a single unit unless you have to. Also be careful when using nukes on the defense, nuking your own territory sucks. Nuking cities should be probably reserved for cities with a powerful garrison and missiles / planes.
When should you use nukes?
If there are just 2 civilizations left, I'm guessing your relation with the remaining one is pretty sour anyway. If you want to achieve victory in any way which isn't by conquering their capital, use the strategic nuking approach. If you want to conquer their capital, save the nukes for the tactical approach. If you are militarily inferior to your opponents you can also save the nuke for defense, but remember to target clumps and avoid targeting your own territory, you will probably need to actively patrol around your continent to find the enemy flotilla before it gets to you.