One of the main things I would want to see is a more historical flow to the game.
Here are some ideas:
1. When you start off building your city, you don't have borders. Improvements can be built on any of the land near the city, but borders come later with advances in technology.
2. Technologies are researched in a different way. People didn't decide: "Hey, let's research agriculture and hope we can build farms someday". I think you should be given points eventually leading to the technology. For example, after your city is built, you have citizens work a nearby patch of grassland. The game will give you a choice if you want to lean towards creating houses for the land (pottery, cottages), or getting food from the land (agriculture, farms). After you choose an option, you start gaining points towards that teachnology, and eventually you gain enough points to have researched the technology. The game can then give you a list of different things to do to get points for new technologies.
Certain tiles would also affect the amount of points you gain per turn: Food tiles like wheat can double the amount of points you get towards agriculture, for example. This would, in my opinion, add a great deal of realism to the game.
3. The way you play affects what your civilization becomes. First, like right now, your civilization starts off with the leader traits. However, depending on your actions, these can be altered. For example, you start as a financial civ. But as you go on you adopt slavery, build cities with lots of population and keep expanding. You are then given a chance (maybe a maximum of two in the game) to change your trait to something that suits you more, like expansive.
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These are just a few ideas, but I think you get the idea of what I'm aiming at. Of course you could make a whole lot more changes following some of these ideas.