This is a good question, one we avoid at the beginning of democracy games because we're too worried about the game becoming one dimensional. In this case, I think it has been a mistake to wait so long to think in terms of victory and what is required to reach it.
Civ4 is markedly different from Civ3 in that chugging along without direction for too long can completely ruin one's position. We don't have the luxury of suddenly building every culture building there is in 100 cities and trying to achieve the 100K culture win, because there is no 100K win. You have to get "legendary culture" in three cities. I'm not sure how the amount of culture to be legendary relates to difficulty levels, but I do know it may be more than 20,000 culture points each.
We have not started early enough to get a culture win. To have any chance at all, at least 3 cities need to be producing the max culture available to them, and we're so far away from that it isn't even funny.
The alternatives are score, military, science, and diplomatic. Taking them in reverse order:
Diplomatic might be fun. It requires we're at least 2nd place in score or build the UN in order to be a candidate for Secretary General, and we have to kiss up to more than half the world's population.
Science, in the form of space race, often ends up in an actual race in Civ4. It is extremely difficult to obtain a lasting tech lead unless you get to the point where your economy can handle double the cities of the opponents. Additionally, as many as 5-7 cities with very high production are needed.
Military via domination is pretty tough without fully developing the infrastructure. As we already see in our current war, the cost of conquest is high. One way out is to keep a lean military, with just enough units to get the job done -- and hope the current opponent doesn't get an alliance which opens a 2nd front.
Military via conquest is a bit easier because you don't have to pay maintenance costs for all the captured cities. Generally movement is slowed down in Civ4, so it takes longer than in Civ3, and the specialized units and promotions make planning very important.
Score is a victory we've never had in a DemoGame, at least not that I remember. We tire of the game too soon for this to be a viable option.
The impact of religion is more economic than cultural. Temples make the people happy, monasteries give a research boost, and cathedrals (etc.) give a boost in culture, happy, and research. (going kinda from memory here, the main idea is that they're all good). Founding a religion and then using a Great Prophet to build the shrine gives +1g for every city which has that religion, foreign or domestic. The religion civics give more benefits.
So where does this all lead us?
- As stated in another thread, we really need to fix the economy.
- If we had been able to found a religion, the benefits would be huge. We're in danger of losing without a religion, and will have to compensate other ways.
- It would seem culture and score are out, leaving military, space, or diplomatic.