The Art of Speed:
1. If you don't need that unit or tech, don't get it. The resources are better spent on something else.
2. If there's any other food available, don't farm.
3. Don't mine any more gold or stone than you need. That means if you're not making much walls/towers or a huge horde of slingers, don't mine much stone. Even gold-heavy strats don't need more than 3-4 gold miners in tool.
4. Some civs are faster than others due to their bonuses. The fast civs are shang, persian, minoan and palmy. The semi-fast civs are roman, yamato and assyrian, phoenie, and sumerian. Last are the "slow civs" - babylon, carth, choson, egypt, greek, hittite and mace. Persian and palmy bonuses are a bit hard to use, the other civs are easier. Shang and roman are especially good if you're just starting out.
5. When picking berries, build your granary right nearby. When cutting wood, put your storage pit nearby.
6. Build houses before you need them. When "before" is depends on how fast you're increasing up your population.