I won't go into specific build orders because they can be difficult to apply and I only really know one*. I'll try to keep advice general.
Scout the area around you early. On a continents or pangea map, a scout should be the first thing you build. Use your scout and warrior to explore the areas around you for ruins and city states - the rewards from these more than pay for the scout.
Try to meet your neighbours with your scouts too. Different leaders have different personalities - as a general rule anyone known for trying to take over the world is likely to do the same in Civilization.**
When you do meet your neighbours, it helps to avoid pissing them off. Even if you do plan to destroy them later, keeping them happy means you can get the advantage later. Things you should ideally avoid doing include***:
- Attacking or bullying city states they are protecting
- Building too many cities (and taking all the land)
- Building cities in land they consider 'theirs'. If it's closer to their capital than it is to yours, or if they were going to build there, it's a bit of a faux pas
If you don't do anything to annoy the AI, they usually won't attack you unless you're exceptionally weak. You don't need a massive army to defend yourself- a few units are enough thanks to the AI's poor unit management. Archers (and their upgrades) are extremely good at defense. Leave one in a city and it can bombard a unit every turn at no risk to itself (until the city is actually taken). Add walls and the city becomes almost impregnable.
*It's my Maya specific build order and I don't want to risk tournament players seeing it in case they point out how bad it is.
** Also Gandhi
*** Human players also don't like me doing these. Just ask @shanodin