If we get the hanging gardens done, I do like the idea of building swords and taking the French out.
Me too!
I am also not sure what role these little independent states play. How do we make them work for us?
How does puppeting work?
OK - my recollection is a bit sketchy, lacking in detail, and confused between vanilla and Gods & Kings, but let me try anyway...
City states are like little independent civs, except that they don't build settlers. They progress at a tech rate ~ on par with probably the top-quartile (can't remember the exact details), and under 1upt (one unit per tile) they are actually pretty good at defending themselves. I have actually seen city states take on other cities states and other civs and put some serious dents in them. Don't underestimate them.
In terms of the game mechanic, the vanilla version city states are a bit bland. Essentially, you do things that they want you to do (which are limited to attacking other city states or barbs, gifting units or gold and perhaps road to them - I can't remember what "quests" are in the vanilla version; its greatly expanded (for the better) in G&K) to earn influence. The more influence you get, the better the rewards. 30 influence makes you "friends", which gives you a minor benefit based on they type of city state (maritime, militaristic etc). Higher friendship levels make you allies etc (note that the city states can only be allied to one civ, and that will be the one with the greatest influence), and allie bonuses are better, including them sharing whatever resources they have with you - and they always have at least one luxury resource.
It is actually entirely possible to ignore city states in games, and playing vanilla I used to largely do this. However, they do have their usefulness (particularly in diplo games). The "problem" is doing enough to keep them happy and maintain freindships (particularly if you don't have a lot of commerce to bribe them, but gifting units can work).
Puppets are different. When you conquer a city, you have the option of razing it, creating a puppet, or Annexing it.
Razing
Razing means that it reduces by 1 population / turn until it disappears, and you suffer unhappiness while that happens.... It also has a diplomatic penalty. You can raze a city that you have captured afterwards, but you cannot raze cities you have founded, nor any capital city.
Annexing
Annexed cities are like Civ4 captured cities - you have full control over them (what they build, what tiles they work etc),
but they add a lot of unhappiness to your civilization. This can be reduced / removed with a courthouse, but they take ages to build and cost a lot in maintenance.
Puppets
Puppets are part of your empire for things like being allies. Buildings they produce contribute to your economy, as does gold and research generated. But they have their own governer who does whatever the hell he / she likes, and you don't get to control them. They don't typically build what you want to build, either. They only build buildings (no wonders, no units), and as such you have to provide units for defense and also workers to improve them. They can be quite frustrating at times, but the benefit is that they don't contribute anything like the unhappiness that an annexed city does (I am pretty sure that you can actually annex them at any time with the click of a button, but once a city is annexed you can't then puppet it).
As a random point to make, there is another factor in capturing or razing cities. One niggling issue that occurs more frequently than you would hope it would. If you have (say) 8 units of aluminium, and build 8 units that require aluminium, you have no aluminium left. If you lose one of your sources of aluminium (say by conquest), all units that rely on it fight at only half strength (IIRC). In capturing cities, there are some
buildings that require aluminium (eg hydro plant, spaceship factory). If you capture a city that has one (or two) or these buildings, then they start using your aluminium... creating a shortage. So your units fight at half strength. Can be really painful when your Rocket Artillery lose all effectiveness, simply because you have captured a city...
By the way, here is a link to an online civilopedia if you don't want to fire-up the game to read the in-built one.
http://www.dndjunkie.com/civilopedia/default.aspx