Civilization – automatically plundering a friendly civilization’s caravan

Whenever you build a Settler, all your extra food goes towards increasing the production of that unit. Due to this your city will always be stagnant when you're building a settler, and I'm willing to bet this is what is happening to you.

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.