It's a bit hard to give too much specific advice without seeing a save game - if you can, post one here....probably around early AD's would be good.
Generally though, it is often better to sacrifice the early settler or two for a Granary, which will provide faster future growth than without it. This will mean you will fall behind the AI in terms of size in the early game, but is usually well worth it as you can grow faster and pump out settlers faster later. Also, early workers are very important, since you want to make sure your tile improvements match your growth. You don't want to be working any unimproved tiles, as that stifles both research and production. One way to assist this is to use reasonably close city spacing so tiles can be shared between towns. If you're having trouble building enough workers, try using an Industrious civ.
Meeting and trading with civs fairly early is also important for keeping up in research. I see you build early warriors for exploration, so you're probably already doing this anyway.
As for wonders, the higher you move up, the more difficult it will be to snare ancient age wonders. If you have your heart set on one, you should start a pre-build of a palace or expensive improvement before you get the wonder-enabling tech and then switch over later. However, wonder building in the early game really just ties up your most productive cities and prevents them from producing what you really need - settlers, workers and defenders. At higher levels, wonder building needs to be very selective. The Pyramids is an obvious choice, but difficult to get at higher levels since it becomes available so early. Don't use entertainers if you can help it. Use the luxury slider to keep them happy, which will be more effective if all your city tiles are improved. Rivers and gold hills are expecially useful for this.
That's just some general thoughts. Post a save or at least a screenshot and someone will have a look and give you some more specific advice.