In the Ancient Wonders scenario, the Oracle can be built from the start and has a different function. Any civ can bring a Great Person within a few tiles of the city with the Oracle to consult it (it gives you a prompt on whether or not you want to consult) - this gives you recon about how the other civs are doing regarding the building of the other wonders (including their current hammer count if applicable). Consulting the Oracle does not consume the Great Person, but a given Great Person can only do it once per game. Technically, you can consult the Oracle even if it is owned by someone else but that doesn't normally happen. I've also never seen the AI send their Great People to consult my Oracle.
The other Wonders have their normal effects but instead of being unlocked by a specific technology, you need to achieve certain milestones (total culture, total GPT accumulated, # of techs, total XP) to be able to build them.
To get this achievement, you just need to get two different Great People in the game and then preferably build the Oracle yourself. Choosing the "Babylonians" (who look like Arabia in the scenario) is probably the easiest since you get a Great Scientist early on. Build the Oracle, then get Honor (Warrior Code) and the free Great General.