The luxuries give you 4 happiness base. More cities/higher population creates unhappiness. If happiness is less than 0 it has bad effects on research/production/gold and unit fighting effectiveness. Having excess happiness gives you golden ages once you get enough, but it's not that important and the amount to reach a gold age goes up every time. You need luxuries to keep your happiness above 0 while expanding and growing your cities and you can also trade them to the ai for around 7 gold per turn.
If you want quick guide for success go babylon and research pottery -> writing, get the free great scientist and use him to build a research tile near your city and make sure it's being worked by the city (click city, click citizen management in top right, select tile to force work). As soon as you do that build the great library, then the national college (make sure to get the tech for that before great library finishes). After national college finishes build 2 or 3 settlers and just focus on building up those cities with workshops, libraries, etc. By turn 100-150 you'll probably wanna start getting walls in your cities and have some military defending so the ai doesn't declare war, but for the most part you want to stay out of war and just tech up and go for either culture victory (fast tech leads to culture buildings, world wonders that all have +culture (especially sistine chapel, cn tower, sydney opera house, maybe others), early archaeology, tourism, etc etc. When you get your first social policy choose tradition and then make sure you have the +15% wonder building by the time you start great library (probably get it second).
Another really quick strat that is nice is go Huns and quick tech their special siege unit, battering ram, and just go on the warpath taking towns. If your warrior gets a weapon upgrade from an ancient ruin he'll become a battering ram and you can take over other players capitals in the first 10 turns. Focus on taking out any expansions as soon as the ai puts them up. Demand tribute from city states that your army is near to. Keep the ai's spread as small as you can, while keeping good cities for yourself and razing the worse ones. Good cities should have good luxury resources nearby and river, some hills or trees... too much for this guide