The "early game" of Civilization is really about learning about your position as best as possible. Every discovery you make whether it's a resource, terrain feature, or where your neighbors are gets you brain thinking about what position you want to get yourself in. The rewards of finding this out early are weighed against the vulnerable position you put yourself in by producing and sending units out to explore and stake claims. Each advantage you acquire, each choice you make early on has an impact on what the borders will look like when it's much more difficult to expand and what you will ultimately need to do in the late game. I think giving this sense to the player is important as well as establishing that you can have an entertaining experience in the late game even if you don't end up on the top militarily early on.
It is already in my game design to have a meaningful early game. But even when you make important decisions, the early game will always be a part of the game where there is very little to do. Civilization fixes this by having exploration, barbarians and ruins to keep you entertained. So i'm exploring some options to have some fun aspects added to the early game. I think both hardcore and casual gamers would enjoy this.
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You could build mines in "remote" locations to supply your cities early on while later on, when you the player has more resources, he can build cities near those resources and remove the mines, it'd mean he'd still have to fight over/claim the resources early on and later on, but seperate those two choices(which resources to go for) so a new player will not have to make choices early on that might cost him his/her late-game, while still allowing for choices that affect the outcome.
I actually quite like the idea of being able to claim territory, without the being-able-to-mine part. This would surely create tension from the beginning of the game and cause some small early game skirmishes.