First off, don't even try and make the other civs like you. They never will! Trust every single one of them, even Gandhi, about as far as you could throw them in person.
Ally, however, with a bunch of city states. By contrast with normal civs, city states
are trustworthy and they can feed you, arm you (a little) and can contribute to around half your culture points. Do their quests for barbarian eradication but NOT conquest of another city state! Build up the gold and buy their influence. Get the Patronage social policy tree early (saving up your points if necessary).
Regarding being attacked, I use the adage "Attack is the best form of defence". Go get them before they get you! Get a bunch of horsemen and (controversially) a few chariot archers. Horsemen are one of the best units in the game: they're available early and are a mix of power and mobility -
and they can move after attacking! Careful use of chariot archers, keeping them out of direct contact but using them only for ranged attacks to soften up targets for the horsemen to come in
on the same turn, earns them lots of promotions and a lot more power than they would seem to have. Get them EARLY and take out the other civs' cities before they're too strong. However, do NOT erase an entire civ. Leave one pathetic city for each and keep an eye on them. (The AI really dislikes civ eradication.) And don't annex or raze the cities but puppet them, at least at first.
There's a lot of basic info in the civ manual, for example about knowing your culture multipliers (Sistine Chapel, Hermitage, Broadcast Towers and Constitution social policy having slingshotted and beelined to Acoustics). However, overall, I concur with erajah's advice: do read the (long) HOF thread. People are getting full cultural victories in this first monthly HOF Gauntlet challenge using Arabia (not the obvious or best choice) at Prince level (good for beginners) and Continents, and some are getting home with victories as early as 1700! You could be next.