How to do things before 3000 bc with the aztecs:
4000 bc: move your settler next to the barbarian village, and found your cirst city. Send the scout you get for free for settling next to barbarians in the most interesting direction. Use 20 of your 25 gold to rush a warrior and capture the barbarian village.About nine times of ten, you get 30-50 gold when capturing your first barbaraian village. Lets assume you get 30 gold, which leaves you with 35 gold. Set your workers to work on two forests.
3900 BC: Use 12 gc to rush your second warrior, and send it towards the barabarian village your scout hopefully discovered. 23 gold left.
3800 BC: Use 12 gc to rush your third warrior and send it in the most interesting direction. 11 gold left.
3700: If your scout, or first or second warrior have either found a mythic place, or barabarian village to conquer, use 12 gc to rush your third warrior.
3600: Now you have four military units roaming the map, and have a very good chance of finding some kind of income in turn 4. Use 12 gc to rush your fourth warrior. Change your production to only food, and use your five military units to explore the map and harvesting gold.
3500: If you have the gold, and there is parts of the map around you that is undiscovered, rush a fifth warrior. If there is not much map to explore, save your 20 gold and aim to gather the 100 nedded to get a free settler.
3100 bc: Your city grows to size 3 and you have visited 9 (1st warrior) + 9 (the scout) + 8 +7 +6+5 = 44 locations on the map. This is normally enough to gather 100 gold.
When playing other civs, just begin with only producing hammers the first five turns. This gives you the first warrior in 3700 bc, and your second warrior in 3500 bc. The chinese, with the luck of 3 nearby forests, will produce 3 warriors in the first five turns.
If you switch to food production in 3500 bc, your city will grow in 3000 bc, just as if you had not managed your workers. The only difference being that you get your first warrior in 3700 bc instead of 3500 bc, and your second warrior in 3500 bc instead of 3000 bc.
Important to remember is that the barbarian villages normally get their second unit in 3500 bc. If you can get to 1-2 barbarians villages before this happens, it will only take you one turn to defeat the barbarians. And with the extra gold, you can rush more military units, or get quicker to 100 gold...
With the aztecs, i have managed to reach 100 gold in 3700 bc, and have fond memories of having two cities, and ten military units by 3000 bc. (Five warriors, two galleys and three scouts)
The aztecs autoheal is fantastic for fast beating of barbarians, since you do not have to heal your units before moving on towards the next barbarian village. The aztecs is at their finest when you reach a barbarian village with 3 units. When you step next to the barb-village, unit 3 will attack you, your unit will autheal after the defensi victory, and be able to defeat three barabrian units in two turns, while getting upgraded to elite.