Caravans and cash are certainly the quickest way to get your wonders built before the AI, but these options may not be available early enough for some of the very early wonders. In order to get at the Great Wall, Pyramids, Colossus, Hanging Gardens, etc. you may not have been able to get to trade, and chances are you don't have large stores of cash to use either.
With these valuable early wonders, my strategy is to simply start early. In my first few cities I'll build a warrior, a settler, another warrior, possibly a temple if on a higher level, and then go for a wonder. Warriors for their speed of building and because they are just as good at martial law, the settlers because they are needed to expand, and the temple to allow the city an extra worker or two before entertainers may be needed.
I know that others will build a second settler, a library, a marketplace or some other combination. However, if you're at Deity level with several other civs, I've always found it best to get going on wonders ASAP and let the later cities build the extra settlers/improvements. On lower levels I may build that extra settler or a phalanx instead of a warrior, but even then it pays to begin early. Still, as Tim mentions, prioritizing is a key issue as well. I'll often give up certain wonders like the Lighthouse or Oracle to be sure I get the ones I want.
Once trade is available, I find that I can get far enough ahead in tech with the early wonders on my side to be able to build more "actual" caravans for trade and build a minimal amount of food caravans for later wonders. The Great Wall means I may not have to build huge defenses right away, the Gardens allow more growth wil less agitation, the Pyramids help with growth in general and keeps the AI from having it, and the Colossus is a great first step for the science city and a perfect "homing" place for those caravans.
------------------
Diplomacy - the art of
saying "Good Doggie"
until you can find a rock